Funkcialaj Ekvacioj, 30 (1987) 169-202 Boundary Estimates for Elliptic Partial Differential Equations in the $ mathscr{L}^{(q, lambda)}$ Spaces of Strong Type By Akira ONO*) (Kyushu University, Japan) Introduction The structures of the $ mathscr{L}^{(q, lambda)}$ spaces were first studied by C. B. Morrey [9] and afterwards by F. John-L. Nirenberg [6]. Motivated by these researches, S. Campanato, G. Stampacchia and others have given general definition of these spaces, which have been investigated by various authors including them (see for example [18], [19] and [21] $)$ . Furthermore, the theory of the spaces has proved to be particularly useful in the study of partial differential equations of elliptic and parabolic type. Researches of elliptic partial differential equations in these spaces were at first made by C. B. Morrey [9], [10] applying his well-known imbedding theorems and afterwards by S. Campanato [3], [4] with the aid of isomorphism theorems and the so-called fundamental inequalities due to him. On the other hand, G. Stampacchia introduced the $ mathscr{L}^{(q, lambda)}$ spaces of strong type [20], the structures of which are more general and complicated than those of the $ mathscr{L}^{(q, lambda)}$ spaces in the usual sense, and greater parts of them were characterized by him, L. C. Piccinini, Y. Furusho, the author and others (see [5], [12]?[14], [16], [17], [19] and [20] $)$ . Furthermore, the author has given in [15] precise interior estimates for solutions of linear uniformly elliptic partial differential equations applying theorems due to S. Agmon-A. Douglis-L. Nirenberg [1], S. Campanato- G. N. Meyers [2], [8], F. John-L. Nirenberg [6], A. Ono-Y. Furusho [17] and the author [12]?[14]. In this paper, the author will deduce apriori estimates near the boundary for solutions of linear uniformly elliptic partial differential equations satisfying general boundary conditions. This article is organized as follows: In §1 relevant definitions, fundamental assumptions on the equations and the first main results, that is, the strong $ mathscr{L}^{(q, lambda)}$ estimates are stated. The characterization of the of certain $ mathscr{L}^{(q, lambda)}$ spaces of strong type is $*)$ This research is partially supported by Grant-in-Aid for Research of Sciences under the Ministry of Education of Japan Government.
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Funkcialaj Ekvacioj, 30 (1987) 169-202
Boundary Estimates for Elliptic Partial DifferentialEquations in the $¥mathscr{L}^{(q,¥lambda)}$ Spaces of Strong Type
By
Akira ONO*)
(Kyushu University, Japan)
Introduction
The structures of the $¥mathscr{L}^{(q,¥lambda)}$ spaces were first studied by C. B. Morrey [9] andafterwards by F. John-L. Nirenberg [6]. Motivated by these researches, S.Campanato, G. Stampacchia and others have given general definition of thesespaces, which have been investigated by various authors including them (see forexample [18], [19] and [21] $)$ .
Furthermore, the theory of the spaces has proved to be particularly useful inthe study of partial differential equations of elliptic and parabolic type.Researches of elliptic partial differential equations in these spaces were at firstmade by C. B. Morrey [9], [10] applying his well-known imbedding theoremsand afterwards by S. Campanato [3], [4] with the aid of isomorphism theoremsand the so-called fundamental inequalities due to him.
On the other hand, G. Stampacchia introduced the $¥mathscr{L}^{(q,¥lambda)}$ spaces of strongtype [20], the structures of which are more general and complicated than those ofthe $¥mathscr{L}^{(q,¥lambda)}$ spaces in the usual sense, and greater parts of them were characterizedby him, L. C. Piccinini, Y. Furusho, the author and others (see [5], [12]?[14],[16], [17], [19] and [20] $)$ .
Furthermore, the author has given in [15] precise interior estimates forsolutions of linear uniformly elliptic partial differential equations applyingtheorems due to S. Agmon-A. Douglis-L. Nirenberg [1], S. Campanato-G. N. Meyers [2], [8], F. John-L. Nirenberg [6], A. Ono-Y. Furusho [17] andthe author [12]?[14].
In this paper, the author will deduce apriori estimates near the boundary forsolutions of linear uniformly elliptic partial differential equations satisfyinggeneral boundary conditions.
This article is organized as follows:In §1 relevant definitions, fundamental assumptions on the equations and
the first main results, that is, the strong $¥mathscr{L}^{(q,¥lambda)}$ estimates are stated.The characterization of the $¥mathrm{t}¥mathrm{r}¥mathrm{a}¥mathrm{c}¥mathrm{e}$ of certain $¥mathscr{L}^{(q,¥lambda)}$ spaces of strong type is
$*)$ This research is partially supported by Grant-in-Aid for Research of Sciences under theMinistry of Education of Japan Government.
170 Akira ONO
given in §2. This enables us to give the estimates up to the boundary in thesespaces.
The strong $¥ovalbox{¥tt¥small REJECT}^{(q,¥lambda)}$ estimates which correspond to the $L^{p}$ estimates in [1] andthe Schauder estimates, that is, the ones in the strong Holder spaces are provedin §3 and §4 respectively.
Main tools for the proof of the theorems in§2?§4 are theorems and techniquesdue to S. Agmon-A. Douglis-L. Nirenberg [1], S. Campanato-G. N. Meyers[2], [8], F. John-L. Nirenberg [6], S. M. Nikol’skii [11], A. Ono-Y. Furusho[17] and the author [12]?[16].
In §5 we apply Morrey-Sobolev type imbedding theorems proved in [14].Namely, at first we prove that the theorems stated in §1 are still valid undermore general conditions. Secondly, precise estimates of the lower order deriva-tives of the solutions are deduced.
Supplementary comments on these theorems are given in §6.
§1. Preliminaries and statement of theorems on the strong $¥mathscr{L}^{(q,¥lambda)}$ estimates
Throughout this paper we denote by $S_{R}$ and $C_{R}$ an arbitrary fixed semi-spherewith radius $R$ in the Euclidean $¥mathrm{w}$ -space $E^{n}$ and its flat boundary, that is, $S_{R}=$
$¥{¥chi=(X_{1},¥cdots, X_{n}):|x|¥leqq R, x_{n}¥geqq 0¥}$ and $C_{R}=S_{R}|_{x_{n}=0}$ .
We always consider subfamilies of real-valued integrable functions on $S_{R}$
and an arbitrary subcube $Q$ of $S_{R}$ with its sides parallel to the axes (from now on,the word “subcube” means such a parallel subcube without loss of generality).Further, we denote the measure of a subcube $Q$ by $|Q|$ and the mean value of a
function $u$ over $Q$ by $u_{Q}$ : $u_{Q}=|Q|^{-1}¥int_{Q}u(x)dx$ .
DeMition 1. A function $u¥in L^{q}(S_{R})$ is said to belong to the space $¥mathscr{L}_{p}^{(q,¥lambda)}(S_{R})$
(the $¥mathscr{L}^{(q,¥lambda)}$ space of strong type $p$), if the following inequality holds for $u$ :
where $1¥leqq p$ , $ q<¥infty$ , $-¥infty<¥lambda<¥infty$ and $¥overline{S}$ is the family of all systems of subcubes$¥{Q_{j}: ¥bigcup_{j}Q_{j}¥subset S_{R}¥}$ of finite number, no two of which have common interior point.
Taking as the norm of the space $¥mathscr{L}_{p}^{(q,¥lambda)}(S_{R})$
Deffiition 2. The space $¥mathscr{L}_{p}^{(q,0)}(S_{¥mathrm{R}})$ is isomorphic to the space $¥mathscr{L}_{p}^{(1,0)}(S_{R})$
Boundary Estimates for Elliptic PDE 171
for any constants $p$ and $q$ such that $ n¥leqq p<¥infty$ and $ 1<q<¥infty$ . Therefore, we callthe space $¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})$ the John-Nirenberg space of strong type $p$ .
DeKition 3. The semi-norm $[v]_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q’¥lambda)}(S_{R})}$ is equivalent to the semi-norm
for any constants $p$ , $q$ and $¥lambda$ such that $ 1<q<¥infty$ , $-q<¥lambda<0$ and $ n/(1-¥alpha)¥leqq p<¥infty$
$(¥alpha=-¥lambda/q)$ . Therefore, we call the space $¥ovalbox{¥tt¥small REJECT}_{p}^{a}(S_{R})$ the space of Holder continuousfunctions of strong type $p$ with exponent $¥alpha$ . Furthermore, we denote by $¥ovalbox{¥tt¥small REJECT}_{p}^{4+a}(S_{R})$
substituting $¥ell+¥alpha$ for $¥alpha$ in the above definition, where $¥ell$ is a fixed positive integer.
Here, we note that the Definitions 2 and 3 are the direct conclusion of thefollowing:
Theorem. The space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda$)$(S_{R})$ is isomorphic to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(1,¥lambda/q^{)}}(S_{R})$
$(¥ovalbox{¥tt¥small REJECT}_{p}^{(-¥lambda/q)} if-q<¥lambda<0)$ and we have
where $x=(x_{1},¥cdots, x_{n})$ and $x^{¥prime}=(x_{1},¥cdots, x_{n1}¥_)$ (from now on, we denote always by $x$
and $x^{¥prime}$$¥mathrm{n}$ -and 1)-dimensional coordinates respectively).
Here, we remark that our method to deduce the boundary estimates is notavailable for the boundary problem of integral form in the case of $l>m_{k}$ (see [1]and §3, §4), and therefore we restrict ourselves to the two problems, namely,$(¥mathrm{E})-(¥mathrm{B})$ and $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ .
Now, we consider the following boundary value problems:
$¥mathrm{I}¥mathrm{I}$ . Complementing condition of the boundary operators relative to $L$ . Theprincipal parts of the polynomials generated by the boundary operators arelinearly independent $Mod.M^{+}(¥mathrm{x}^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$. This means that the polynomials in $¥xi_{n}$ :
are linearly independent $Mod.M^{+}(¥chi^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$ . Namely, let the polynomial$¥sum_{i=1}^{m}b_{i,k}(x^{¥prime}, ¥xi^{¥prime})¥xi_{n}^{i-1}$ be the remainder when the polynomial $¥mathrm{B}k’(¥chi^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$ is dividedby $M^{-¥vdash}(x^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$ . Then there exists a positive constant $c$ such that the followinginequality holds:
where $l$ $¥geqq l_{1}=(2m, m_{k}+1)$, $1<p$ , $ q<¥infty$ , $0<¥lambda<n$ and $1/p<a=n/p-¥lambda/q<$$1^{2)}$.
And, for the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ :$(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}}$,
I. Condition on L. $L$ is uniformly elliptic. The uniform ellipticity in thiscase means that there exists a constant $E$ greater than unity such that the followinginequality holds:
We denote $¥{¥xi_{n,k}^{+}(x^{¥prime}, ¥xi^{¥prime})¥}_{k=1,m}¥ldots$, by the same manner as before and define $M^{+}(x^{¥prime}$ ,$¥xi^{¥prime}$ , $¥xi_{n})$ by (1.4).
?
2) We always denote $(¥mathit{2}m, m_{k}+1)$ and $n/p-¥lambda/q$ by $l_{1}$ and $a$ respectively.
174 Akira ONO
$¥mathrm{I}¥mathrm{I}$ . Complementing condition of the boundary operators relative to $L$ . Weassume that the principal parts of the polynomials in $¥xi_{n}$ generated by the boundaryoperators:
where $¥ell<(¥mathit{2}m, m_{k}+1)$ and greater than the maximum order of $x_{n}$ differenti-ation occuring in the $B_{k}$ , are linearly independent $Mod.M^{+}(x^{¥prime}, ¥xi^{¥prime}, ¥xi_{n})$.
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$ . The coefficients and functions $¥{a_{¥beta¥gamma},f_{¥gamma}¥};¥{b_{k,¥beta,¥gamma^{¥prime}}, g_{k,¥gamma^{¥prime}}¥}$ are smooth.Namely, we assume
where $1<p$, $ q<¥infty$ , $0<¥lambda<n$ and $1/p<a=n/p-¥lambda/q<1$ .
Now, our first main results read as follows:
Theorem 1. Let $u$ be an $H^{¥mathrm{A}_{1},p_{-}}$solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}$ .
Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq¥beta}$ belong to the space $g_{p}^{(q,¥lambda)}(S_{R})$ andthe following estimate holds for $u$ :
Theorem 2. Let $u$ be an $H^{¥mathit{4},p}$ -solution of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}},$ .
Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}}$ belong to the space $ g_{p}^{(q},¥lambda$ ) $(S_{R})$ andthe following estimate holds for $u$ :
where $1<p$ , $ q<¥infty$ , $0<¥lambda<n$ , $1/p<a=n/p-¥lambda/q<1$ and $C$ is a constant inde-pendent offunctions $v$ and $w$ .
2. Conversely, let $w$ be a function belonging to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(C_{R})$ .
Then, there exists a function $v$ such that $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$ and$v|_{x_{n}=0}=w$ . Moreover, we can extend the function $w$ so that the following esti-mate holds:
where $ 1<p<¥infty$ , $ 0<b-1/p¥neq$ integer and $C$ is a constant independent offunctions$v$ and $w$ .
2. Conversely, let $w$ be a function belonging to the space Lip $(b-1/p,$ $p$ ,$C_{R})$ . Then, there exists a function $v$ such that $v$ belongs to the space Lip $(b,$ $p$ ,$S_{R})$ and $v|_{x_{n}=0}=w$ . Moreover, we can extend the function $w$ so that the followingestimate holds:
Proof of Theorem 3.We give the proof of the cases 1 and 2 simultaneously. By Lemma 1, the
spaces $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$ and $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,)}¥lambda(C_{R})$ are isomorphic to the space Lip $(a, p, S_{R})$ andLip $(a -1/p, p, C_{R})$ respectively and we have
Secondly, we prove the following theorem which plays an important rolefor the proof of the Schauder estimates:
Theorem 4.1. Let v be a function belonging to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{4+¥alpha}(S_{R})$. Then, there
Boundarv Estimates for Elliptic PDE 177
exists a $f¥dot{u}nct¥iota ¥mathit{9}$on $w$ such that $w$ belongs to the spac $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+¥alpha}(C_{R})$ and $v|_{¥mathrm{x}_{n}=0}=w$ .Moreover, we have
where $l$ is a non-negative integer, $0<¥alpha<1$ and $ n/(1-¥alpha)¥leqq p<¥infty$ .2. Conversely, let $w$ be a function belonging to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+¥alpha}(C_{R})$ .
Then, there exists a function $¥iota$
. such that $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{l+¥alpha}(S_{R})$ and$v|_{x_{n}=0}=w$ . Moreover, we can extend the function $wo‘ O$ that the following esti-mate holds:
where $¥ell$ , $¥alpha$ and $p$ are as in Theorem 4.
Proof of Theorem 4. By Lemma 3 the spaces $¥ovalbox{¥tt¥small REJECT}_{p}^{D+a}(S_{R})$ and $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{P}+a}(C_{R})$ areisomorphic to the spaces Lip $(l+¥alpha+n/p, p, S_{R})$ and Lip $(l+¥alpha+(n-1)/p, p, C_{R})$
respectively. Applying Lemma 2, the conclusion is immediate.
Here, we make the following important:
Remark 1. The norm of the space $¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})$ has been defined in Definition1 and we observe that the solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ or $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ underthe assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}$ or $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(q¥lambda)}}$, means $H^{¥mathrm{A}_{1},q_{-}}$ or $¥mathrm{H}^{¥mathrm{A},¥mathrm{q}}$-solution. However,by Lemmas 1, 3 and the procedure of the proofs of the theorems in §3 and §4,we can understand that the functions $¥{/, g_{k}¥}$ or $¥{f_{¥gamma}, g_{k,¥gamma^{¥prime}}¥}$ satisfy smoothnessconditions in certain subspace of the $L^{p}$ space.
Therefore, we always consider the solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ or theProblem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ in the $L^{p}$ framework.
§3. Proof of Theorems 1 and 2
For the proof of Theorem 1, we prepare the following:
Lemma 4 ([1] Agmon-Douglis-Nirenberg). Let $u$ be an $H^{4_{1},p}$ -solution $(¥ell_{1}$
178 Akira ONO
$=(¥mathit{2}m, m_{k}+1))$ of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishing near the curved boundaryof $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q}.¥lambda)}}$ I-III.
Then, in fact $u$ belongs to the space $H^{4,p}(S_{R})$ and we have
where $||g_{k}||_{¥mathrm{A}m_{k^{¥_}}1/p}¥_=¥inf_{v|_{x_{n}=0}=g_{k}(x^{¥prime})}||v||_{H^{p-m_{k},p}(S_{R})}^{3)}$.
Now, we are going to give the
Proof of Theorem 1.
From $(¥mathrm{E})-(¥mathrm{B})$ , we can easily verify that the following equalities hold:$¥sum_{|¥beta|¥leqq 2m}a_{¥beta}(x)D^{¥beta}¥{u(x+h)-u(x)¥}$
3) We always denote $¥inf$ $¥{_{1}|v||_{H}¥mathrm{A}_{p(},s_{R)};v(¥chi)_{x_{n}=0}^{¥mathrm{I}}=g(x^{/})¥}$ by $||g||_{¥mathrm{P}1/p}¥_$ according to [1] ($l$ issometimes replaced by other positive integers).
Here, we note that all of the terms which appear in the right hand side are finiteby the condition $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥lambda)}}$, and Lemma 4. Furthermore, we have by Lemma 4
This means that the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq A}$ belong to the space Lip $(a, p, S_{R})$ andtherefore to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,)}¥lambda(S_{R})$ by Lemma 1.
Hence, we can finally conclude that the following estimate holds for $u$ :
Next, by similar calculations to those of the proof of Theorem 1, we give the
Proof of Theorem 2.
For this purpose, we need the following lemma instead of Lemma 4:
Lemma 5 ([1] Agmon-Douglis-Nirenberg). Let $u$ be an $H^{0,p_{-}}$solution( $l$ $<(¥mathit{2}m, m_{k}+1)$ and greater than the maximum order of $x_{n}$ differentiationoccuring in the $B_{k}$) of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishing near the curved boundaryof $S_{R}$ under the assuption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}}$, I-III.
Here, we note that all of the terms which appear in the right hand side arefinite by the condition $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(_{q}.¥lambda)}}¥mathrm{I}¥mathrm{I}¥mathrm{I}$ and Lemma 5. Furthermore, we have byLemma 5:
This means that the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq¥ell}$ blong to the space Lip $(a, p, S_{R})$
and therefore to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda$) $(S_{R})$ by Lemma 1.
Hence, the proof of Theorem 2 is complete.
§4. Schauder estimate
At first, we make the following assumptions in place of the assumptions$(A)_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(¥lambda)}}$, and $(A)_{¥acute{e}_{p}^{(q¥lambda)}}$, for the Problems $(¥mathrm{E})-(¥mathrm{B})$ and $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ respectively.
$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{a}}$
$¥mathrm{I}$ , $¥mathrm{I}¥mathrm{I}$ : Same as the conditions (A)$e_{p^{q}}^{(,¥lambda)}¥mathrm{I}$ , $¥mathrm{I}¥mathrm{I}$ .Ill: The coefficients and functions $¥{a_{¥beta}, f¥};¥{b_{k,¥beta}, g_{k}¥}$ belong to the spaces
$¥ovalbox{¥tt¥small REJECT}_{p}^{A-2m+¥alpha}(S_{R})$ and $¥ovalbox{¥tt¥small REJECT}_{p}^{¥rho-m_{k}+¥alpha}(C_{R})$ respectively, where $l¥geqq l_{1}=(2m, m_{k}+1)$,$0<¥alpha<1$ and $ n/(1-¥alpha)¥leqq p<¥infty$ .
$¥mathrm{I}$ , $¥mathrm{I}¥mathrm{I}$ ; Same as the conditions $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}},¥mathrm{I}$ , $¥mathrm{I}¥mathrm{I}$ .Ill: The coefficients and functions $¥{a_{¥beta¥gamma},f_{¥gamma}¥};¥{b_{k,¥beta,¥gamma},, g_{k,¥gamma^{¥prime}}¥}$ belong to the
spaces $¥ovalbox{¥tt¥small REJECT}_{p}^{a}(S_{R})$ and $¥ovalbox{¥tt¥small REJECT}_{p}^{1+¥alpha}(C_{R})$ respectively, where $0<¥alpha<1$ and $ n/(1-¥alpha)¥leqq p<¥infty$ .Then, the main results in this section read as follows:
Theorem 5. Let $u$ be a $C^{¥mathrm{A}_{1}+¥alpha_{-}}$solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{a}}$ .
184 Akira ONO
Then, $u$ is in fact $¥ovalbox{¥tt¥small REJECT}_{p}^{A+¥alpha_{-}}$solution and the following estimate holds for $u$ :
Theorem $¥epsilon$. Let $u$ be a $C^{A+¥alpha_{-}}$solution of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{a}}$.
Then, $u$ is in fact $¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}+¥alpha_{-}}$solution and the following estimate holds for $u$ :
Proof of Theorem 5. For this purpose, in addition to Theorem 4 and Lemma3 we need the following lemma instead of Lemma 4:
Lemma 6 ([1] Agmon-Douglis-Nirenberg). Let $u$ be a $C^{¥mathrm{A}_{1}+a_{-}}$ solution ofthe Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishing near the curved boundary of $S_{R}$ under the as-sumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{a}}$ .
obviously $||f||_{C^{¥mathrm{A}- 2m+a}(S_{R})}$ and $¥{||g_{k}||_{C^{flm_{k}+¥alpha}(C_{R})}¥_¥}$ are majorized by $||f||_{¥ovalbox{¥tt¥small REJECT}_{p}^{p_{-2m+a}}}(S_{R})$
and $¥{||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{p_{-m_{k}+a}}(C_{R})}¥}$ respectively and therefore we obtain the followinginequality:
Applying Lemma 3 to the left hand side, the proof of Theorem 5 is complete.
Next, we give the proof of Theorem 6 with the aid of Theorem 4, Lemma 3and the following:
Lemma 7 ([1] Agmon-Douglis-Nirenberg). Let $u$ be a $C^{¥mathrm{A}+¥alpha_{-}}$solution $(t$ $<$
$(¥mathit{2}m, m_{k}+1)$ and greater than the maximum order of $x_{n}$ differentiationoccuring in the $B_{k}$) of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishing near the curved boundary
of $S_{R}$ under the assumption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{a}}$ .
Applying Lemma 3 to the left hand side, the proof of this theorem is complete.
§5. Applications of Morrey-Sobolev type imbedding theorems
We have proved in [14] the following:
Lemma 8. Let $v$ be a function such that the derivatives $¥{v_{X}¥}$ belon $¥backslash q$ to thespace $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,¥lambda)}(S_{R})$ , where $p$ , $q$ and $¥lambda$ are constants such that $1<p$, $ q<¥infty$ , $0<¥lambda<n$
and $n/p<¥lambda/q$ .Then, the following estimates hold for $v$ :1. $ q<¥lambda$ and $¥lambda/¥tilde{q}<n/p<¥lambda/q;v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(_{¥tilde{q},¥lambda})}(S_{R})$ and we have
2. $ q=¥lambda$ ; $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})$ and we have
4) Throughout the remainder of this paper, we always denote by $¥tilde{q}$ the constant defined here.
188 Akira ONO
(5.2) $[v]_{¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})}¥leqq the$ right hand side of (5.1)
3. $ q>¥lambda$ ; $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{1-¥lambda/q}(S_{R})$ and we have
(5.3) $[v]_{¥ovalbox{¥tt¥small REJECT}_{p}^{1-¥lambda/q}}(S_{R})¥leqq the$ right hand side of (5. 1).
Now, our first main result in this section which is deduced with the aid ofthis lemma reads as follows:
Theorem 7. Theorems 1 and 2 are still valid under the assumptions$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥lambda)}}$, and $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p^{q}}^{(¥lambda)}},$ , even if we replace the condition “ $1/p<a=n/p-¥lambda/q<1$ ”
by the condition “$0<a=n/p-¥lambda/q<1/p$ ’ ’.Proof of Theorem $1^{5)}’$.
By a similar procedure as in the proof of Theorem 1, we have
$¥leqq v(x)|_{x}$ in $¥mathrm{o}^{=g_{k}(¥mathrm{x}^{¥prime})}¥mathrm{f}||v(x)||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathrm{P}-m_{k}+a,p,S_{R})}$
applying Lemma 2, this is$¥leqq C||g_{k}||_{¥mathrm{L}¥mathrm{i}¥mathrm{p}(¥mathrm{A}-m_{k}+a-1/p,p,C_{R})}$
and with the aid of Lemma 1, we have
$¥leqq C¥sum_{|¥gamma^{¥prime}|¥leqq ¥mathrm{A}-m_{k}-1}||D_{x^{¥prime}}^{¥gamma^{¥prime}}g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(¥tilde{q},¥lambda)}(C_{R})}$in the case of $q<$ ).
or
$¥leqq C¥sum_{|¥gamma^{¥prime}|¥leqq D-m_{k}-1}||D_{x}^{¥mathcal{Y}^{¥prime}},g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(C_{R})}$in the case of $ q=¥lambda$
$¥leqq C||g_{k}||_{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-m¥mathrm{k}^{-j/q}}(C_{R})}$ in the case of $ q>¥lambda$
For the remainder of the proof of this theorem, we make a similar argumentas in the proof of Theorem 1, and we obtain finally the following estimate for $u$ :
This means that the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}}$ belong to the space Lip $(a, p, S_{R})$
and therefore we can conclude that $¥sum_{|¥beta|¥leqq ¥mathrm{A}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}(S_{R})}$ is majorized by theright hand side of above inequality applying Lemma 1.
This completes the proof of Theorem 1’.
Proof of Theorem 2’.By a similar procedure as in the proof of Theorem 2, we have
For the remainder of the proof of this theorem, we apply Lemmas 1 and 5 tothe left hand side and $||u||_{H^{p,p}(S_{R})}$ respectively, and we have finally the followingestimate for $u$ :
Here, combining Theorems 1.1’ and $2,2^{¥prime}$ , we can state the theorems on thestrong $¥ovalbox{¥tt¥small REJECT}^{(_{q,¥lambda})}$ estimates in the unified forms.
192 Akira ONO
Theorem A. Let $u$ be an $H^{¥mathrm{A}_{1},p_{-}}$solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}$ .
Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}}$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda$) $(S_{R})$ andthe following estimate holds for $u$ :
where $¥ell¥geqq¥ell_{1}=(2m, m_{k}+1)$, $1<p$ , $ q<¥infty$ , $0<¥lambda<n$ , $0<_{¥backslash }a=n/p-¥lambda/q<1$ ,$a¥neq 1/p$ and $C$ is a constant independent of $u$ .
Theorem A. Let $u$ be an $H^{4,p_{-}}$solution of the Problem $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}(l<$
$(¥mathit{2}m, m_{k}+1)$ and greater than the maximum order $o.fx_{n}$ differentiationoccuring in the $B_{k}$) vanishing near the curved boundary of $S_{R}$ under the as-sumption $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(_{q},¥lambda)}}$ .
Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}}$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{7(q,¥lambda)}(S_{R})$ andthe following estimate holds for $u$ :
where $1<p$ , $ q<¥infty$ , $0<¥lambda<n$ , $0<a=n/p-¥lambda/q<1$ , $a¥neq 1/p$ and $C$ is a constantindependent of $u$ .
Now, concerning the exceptional case, that is, the case of $a=n/p-¥lambda/q=1_{/}^{l}p$,
we can deduce a proposition analogous to Theorems A and $¥mathrm{B}$ under slightlystronger conditions.
Proposition 1. Let $u$ be an $H^{¥mathrm{A}_{1},p_{-}}or$ $H^{p_{p_{-}}}$,solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$
or $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ vanishing near the curved boundary of $S_{R}$ under the assumption$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(q¥lambda)}}$, or $(A)_{¥acute{¥ovalbox{¥tt¥small REJECT}}_{p}^{(_{q},¥lambda)}}$ respectively with $a=n/p-¥lambda/q=1/p$ . Furthermore, weassume
$g_{k}¥in H^{¥Lambda-m_{k},p}(C_{R})$ $(t ¥geqq t_{1}=(2m, m_{k}+1))$ for $(¥mathrm{E})-(¥mathrm{B})$
or
$g_{k,¥gamma^{¥prime}}¥in H^{1,p}(C_{R})$ ( $¥ell<(¥mathit{2}m, m_{k}+1)$ and greater than the
Boundary Estimates for Elliptic PDE 193
maximum order of $x_{n}$ differentiation occuring in the $B_{k}$) for $(E)^{¥prime}-(¥mathrm{B})^{¥prime}$ .
Then, in fact the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq^{p}}$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(q,)}¥lambda(S_{R})$ andwe have
for $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$ .
Applying Lemma 1 to the left hand side, the proof of this proposition iscomplete.
Next, we prove the following:
Theorem 8. Let $u$ be an $H^{¥mathrm{A}_{1},p_{-}}$solution $(l_{1}=(2m, m_{k}+1))$ of the Pro-blem $(¥mathrm{E})-(¥mathrm{B})$ vanishing near the curved boundary of $S_{R}$ under the assumption$(A)_{¥ovalbox{¥tt¥small REJECT}_{p}^{(_{q},¥lambda)}}$ . Furthermore, we assume that $l$ is greater than $l_{1}$ .
Then, the following estimates hold for $u$ :1. $ q<¥lambda$ , $ nq/¥lambda<p<(n-1)¥tilde{q}/¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq A-1}$ belong to the
space $¥ovalbox{¥tt¥small REJECT}_{p^{¥tilde{q},¥lambda}}^{()}(S_{R})$ and we have
which completes the proof of this case.By similar arguments as in the proof of the case 1, we can easily deduce the
estimates (5. 10) and (5.11).Hence, the proof of this theorem is complete.
Furthermore, we have proved in [14] the following:
Lemma 10. Let $v$ be a function such that the derivatives $¥{v_{x}¥}$ belong to thespace $¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda$
)$(S_{R})$ , where $p$ , $q$ and $¥lambda$ are constants such that $1<p$ , $ q<¥infty$ , $0<¥lambda<n$
and $0<a=n/p-¥lambda/q<1$ .
Then, the following estimates hold for $v$ :1. $ q<¥lambda$ ; $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{(_{¥tilde{q},¥lambda})}(S_{R})$ and we have
where $r$ is an arbitrary constant greater than $ nq/¥lambda$ .2. $ q=¥lambda$ ; $v$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p}^{(1,0)}(S_{R})$ and we have
(5. 13) $[v]_{¥Psi_{r}^{(1,0)}(S_{R})}¥leqq the$ right hand side of (5. 12)
where $r$ is as in 1.3. $ q>¥lambda$ ; $v$ belongs to tfie space $¥ovalbox{¥tt¥small REJECT}_{r}^{1-¥lambda/_{q}}(S_{R})$ and we have
(5. 14) $[v]_{¥ovalbox{¥tt¥small REJECT}_{(S_{R})}^{1-¥lambda/q}},¥leqq the$ right hand side of (5.12)
where $r$ is as in 1.
Now, our last main results in this section read as follows:
Theorem 9. Let $u$ be a solution as in Theorem $A$ .
196 Akira ONO
Then, the following estimates hold for $u$ :1. $ q<¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{P}1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{r^{¥tilde{q},¥lambda}}^{()}(S_{R})$ and
where $r$ is an arbitrary constant greater than $ nq/¥lambda$ .
2. $ q=¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq D1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})$ andwe have
(5. 16) $¥sum_{|¥beta|¥leqq p-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})}¥leqq the$ right hand side of (5.15)
where $r$ is as in 1.3. $ q>¥lambda$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{D-¥lambda ¥mathit{1}q}(S_{R})$ and we have
(5.17) $||u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{p_{-¥lambda/q}}}(S_{R})¥leqq the$ right hand side of (5. 15)
where $r$ is as in 1.
Theorem 10. Let $u$ be a solution as in Theorem $B$ .
Then, the following estimates hold for $u$ :1. $ q<¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}1}¥_$ belong to the space $g_{r^{¥tilde{q},¥lambda}}^{()}(S_{R})$ and
where $r$ is an arbitrary constant greater than $ nq/¥lambda$ .
2. $ q=¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq A1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})$ andwe have
(5. 19) $¥sum_{|¥beta|¥leqq^{p-1}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})}¥leqq the$right hand side of (5. 18)
where $r$ is as in 1.3. $ q>¥lambda$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{4-¥lambda/q}(S_{R})$ and we have
(5.20) $||u||_{¥ovalbox{¥tt¥small REJECT}_{¥mathrm{r}}^{4-¥lambda/q_{(S_{R}¥rangle}}}¥leqq the$ right hand side of (5. 18)
Boundary Estimates for Elliptic PDE 197
where $r$ is as $¥dot{l}n1$ .
We give simultaneously the
Proof of Theorems 9 and 10. With the aid of Theorems A and $¥mathrm{B}$ , we have
$¥sum_{|¥beta|¥leqq ¥mathrm{A}}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{q}}^{(,¥lambda)}(S_{R})}¥leqq$ the right hand side of (5. 15) or (5.18)
By taking $¥{v_{X}¥}=¥{D^{¥beta}u¥}_{|¥beta|=¥mathrm{A}}$ in Lemma 10, the conclusion is immediate.
Here, we note that we can easily prove with the aid of Lemma 10 the fol-lowing:
Proposition 2. Let $u$ be a solution as in Proposition 1.Then, the following estimates hold for $u$ :1. $ q<¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{(_{¥tilde{q},¥lambda})}(S_{R})$ and
$+¥sum_{k=1}^{m}|¥gamma^{¥prime}¥sum_{|¥leqq m_{¥mathrm{k}}-p+1}||g_{k,¥gamma^{¥prime}}||_{H^{1,p}(C_{R})}+||u||_{L^{p}(S_{R})}¥}$ for $(¥mathrm{E})^{¥prime}-(¥mathrm{B})^{¥prime}$
where $r$ is an arbitrary constant greater than $ nq/¥lambda$ .
2. $ q=¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}1}¥_$ belong to the space $g_{r}^{(1,0)}(S_{R})$ andwe have
(5.23) $¥sum_{|¥beta|¥leqq ¥mathrm{P}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{(1,0)}(S_{R})}¥leqq the$right hand side of (5.21) or (5.22)
where $r$ is as in 1.3. $ q>¥lambda$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{r}^{p-¥lambda/q}(S_{R})$ and we have
(5.24) $||u||_{¥ovalbox{¥tt¥small REJECT}_{r}^{¥mathrm{P}-¥lambda/q}}(S_{R})¥leqq the$ right hand side of (5.21) or (5.22)
where $r$ is as in 1.
We terminate this section by proving the following proposition concerninganother exceptional case.
198 Akira ONO
Proposition 3. Let $u$ be an $H^{A_{1},p_{-}}$solution of the Problem $(¥mathrm{E})-(¥mathrm{B})$ vanishingnear the curved boundary of $S_{R}$ under the assumption $(A)_{¥ovalbox{¥tt¥small REJECT}_{p^{q¥prime}}^{(¥lambda)}}$ , where $ 1<q<¥infty$ ,$0<¥lambda<n$ and $ p=nq/¥lambda$ . Furthermore, we assume that 1 is greater than $¥ell_{1}$ .
Then, the estimates of the same type as in Theorem 9 hold for $u$ .
For the proof, we need the following:
Lemma 11 ([20] Stampacchia). Let $r$ and $r^{¥prime}$ be arbitrary constants satisfy-ing $ 1<r^{¥prime}<r<¥infty$ .
Hence, the proof of this case is complete applying Lemma 11 to the right handside of the above inequality.
By similar arguments as in the proof of the case 1, we can deduce the esti-mates as in Theorem 8 for the cases 2 and 3.
This completes the proof of this proposition.
Here, we make the following:
Remark 2. This proposition asserts that Theorem 9 is valid as long as$0¥leqq a=n/p-¥lambda/q<1_{;}a¥neq 1/p$ .
6. Comments on the theorems
1. According to Campanato [3], we make the following:
Definition $¥epsilon$. Let $X$ be a normed function space. Then, a function $¥zeta$ issaid to belong to the multiplicator space on $X:M(X)$, if the following inequalityholds for an arbitrary function $v$ belonging to the space $X$ :
(6. 1) $||¥zeta v||_{X}¥leqq C||v||_{X}$
Boundary Estimates $J¥dot{o}r$ Elliptic PDE 199
where $C$ is a constant independent of $v$ .
Therefore, we take fairly wide spaces as the multiplicator spaces so as todeduce the preceding theorems as follow:
Theorems $¥mathrm{A}$ , 8 and 9: $X=¥{v;¥{D^{¥beta}v¥}_{|¥beta|¥leqq p}¥in¥ovalbox{¥tt¥small REJECT}_{p}^{(q},¥lambda)(S_{R})¥}$ ;
Here, we make the following remarks on the Schauder estimates.
Remark 3. We have assumed that the solution $u$ belongs to the spaces$C^{p_{1}+¥alpha}(S_{R})$ and $C^{¥mathrm{A}+a}(S_{R})$ respectively so as to deduce the strong Holder estimatesup to the boundary.
On the other hand, we may assume that $u$ belongs only to the spaces $H^{2m,p}(S_{R})$
and $H^{p_{p}},(S_{R})$ respectively for the interior Schauder estimates. We refer [1] and[15].
Remark 4. If $-q<¥lambda<0$ in Theorems A and $¥mathrm{B}$ , then we may set $¥alpha=-¥lambda/q$
$(0<¥alpha<1)$ and therefore $M(X)=¥{c^{p-2m+a+n}/p(S_{R}), c^{l-m_{k}+¥alpha+n}/p(C_{R})¥}$ and$¥{c^{¥mathrm{a}+n}/p(S_{R}), C^{1+¥alpha+n/p}(C_{R})¥}$ respectively.
On the other hand, in Theorems 5 and 6 $M(X)=$ $¥{¥ovalbox{¥tt¥small REJECT}_{p}^{¥mathrm{A}-2m+a}(S_{R}), ¥ovalbox{¥tt¥small REJECT}_{p}^{l-m_{¥mathrm{k}}+¥alpha}(C_{R})¥}$
and $¥{¥ovalbox{¥tt¥small REJECT}_{p}^{¥alpha}(S_{R}), ¥ovalbox{¥tt¥small REJECT}_{p}^{1+¥alpha}(C_{R})¥}$ which are isomorphic to the spaces {Lip $(t$ $-2m+¥alpha+$
$¥alpha+(n-1)/p$ , $p$ , $C_{R})¥}$ respectively by Lemma 3. Obviously, these spaces arewider than the corresponding multiplicator spaces in Theorems A and B.
Therefore, Theorems 5 and 6 give more precise estimates for negative $¥lambda$ thanTheorems A and $¥mathrm{B}$ respectively.
2. Throughout this paper, we have restricted ourselves to the case of $ 0¥leqq$
$a=n/p-¥lambda/q<1$ .
Here we make the following:
Remark 5. If $a=n/p-¥lambda/q$ is equal to unity, we can get “genuine” Sobolevestimates for some of the preceding theorems as follow:
$2(¥mathrm{X})$ Akira ONO
Theormes 1 and 2: The solution $u$ belongs to the Sobolev space $H^{¥mathrm{A}+1,p}(S_{R})$
where $1<p$ , $ q<¥infty$ , $-q<¥lambda<n$ and $¥lambda/q=n/p-1$
In particular, the space $¥ovalbox{¥tt¥small REJECT}_{p}^{¥alpha}(S_{R})$ is isomorphic to the Sobolev space $H^{1,p}(S_{R})$ ,where $0<¥alpha<1$ and $p=n/(1-¥alpha)$ .
Theorems 9 and 10:
Boundary Estimates for Elliptic PDE 201
1. $ q<¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq^{p}}¥_ 1$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{(_{¥tilde{q},¥lambda})}(S_{R})$ andwe have
(6.5) $¥sum_{|¥beta|¥leqq A-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{(¥tilde{q},¥lambda)}(S_{R})}¥leqq$ the right hand side of (6.2)
where $1<p<n$, $0<¥lambda<n$ and $p^{*}$ is the Sobolev’s exponent: $1/p^{*}=1/p-1/n=$
$¥lambda/nq$ .
2. $ q=¥lambda$ ; the derivatives $¥{D^{¥beta}u¥}_{|¥beta|¥leqq ¥mathrm{A}1}¥_$ belong to the space $¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{(1,0)}(S_{R})$ andwe have
(6.6) $¥sum_{|¥beta|¥leqq ¥mathrm{A}-1}||D^{¥beta}u||_{¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{(1,0)}(S_{R})}¥leqq$ the right hand side of (6.2)
where $p$ , $¥lambda$ , $q$ and $p^{*}$ are as in 1.3. $ q>¥lambda$ ; $u$ belongs to the space $¥ovalbox{¥tt¥small REJECT}_{p^{*}}^{¥mathrm{A}-¥lambda/q}(S_{R})$ and we have
(6.7) $||u||_{¥ovalbox{¥tt¥small REJECT}(S_{R})}p^{*}4-¥lambda/q¥leqq$ the right hand side of (6.2)
where $p$ , $¥lambda$ , $q$ and $p^{*}$ are as in 1.
For the proof of this remark, we refer Lemma 1, the condition $ p^{*}=nq/¥lambda$ ,Lemma 9 and Sobolev’s lemma.
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nuna adreso:Department of MathematicsCollege of General EducationKyushu UniversityFukuoka 810, Japan