On the regularity of solutions to the Poisson equation in Orlicz-spaces

Transcription

1 On the regularity of solutions to the Poisson equation in Orlicz-spaces E. Azroul A. Benkirane M. Tienari Abstract We prove in this paper some regularity results of solutions of the Poisson equation u = f, in Orlicz spaces. Introduction Let Ω be an open subset of R N with N 2andletf be a distribution on Ω. We consider the Poisson equation u = f in Ω. (E) The regularity of solutions u of equation (E), in relation to the regularity of the second member f is one of the classical questions concerning this equation (cf.[4-5]). In particular, it s well be known that: (R ) If f is a distribution of order (resp. 0), then the solution u L p loc(ω) for all p< N (resp. u N Lp loc (Ω) for all p< N u and L q N loc (Ω) for all q< ). N (R 2 ) If f L r loc(ω) with r> N 2 continuously differentiable) on Ω. (resp. r>n), then u is continuous (resp. Received by the editors May 998. Communicated by J. Mawhin. 99 Mathematics Subject Classification : 35J05, 35D0. Key words and phrases : N-functions, Orlicz spaces, Distributions, Poisson equation, Newtonian potential. Bull. Belg. Math. Soc. 7 (2000), 2

2 2 E. Azroul A. Benkirane M. Tienari (R 3 ) If f L p (Ω), then 2 u L p (Ω) for all <p<. When p = (resp. p = ), the second derivatives of u are in general not integrable (resp. bounded) in Ω. In the case where f lies in an Orlicz space L A (Ω), the result (R 3 ) is proved in [2] under the assumption that the N-function A and its conjugate A satisfy the 2 condition. It s our purpose in the first part of this paper, to prove the analogous regularity results of (R )and(r 2 ) in the general setting of Orlicz-spaces. The results obtained here constitute in particular a refinement of the L p case. The second part is devoted to obtain in the radial case some regularity results on the second derivatives u, without using any assumptions on the N-function A. 2 Preliminaries In this section we list some definitions and well-known facts about N-functions, Orlicz spaces and Newtonian potential. For more details, we refer the reader to [ ]. 2.- Let A : R + := [0, ) R + be an N-function, i.e. A is continuous, convex, A(t) with A(t) > 0fort>0, 0 (resp. + ) ast 0 + (resp. t + ). t Equivalently, A admits the representation: A(t) = t 0 a(s)ds, where a : R + R + is nondecreasing, right continuous, with a(0) = 0, a(t) > 0for t>0anda(t) + as t +. The N-function A conjugate to A is defined by: A(t) = t 0 a(s)ds, where, a(t) =sup{s, a(s) t}. Clearly A = A and one has Young s inequality: t.s A(t)+A(s), for all s, t > 0. It s well known that we can assume that a and ā are continuous and strictly increasing. We will extend the N-functions into even functions on all R. The N-function A is said to satisfy the 2 condition (resp. near infinity) if there exists k>0 (resp. t 0 > 0) such that A(2t) ka(t), for all t 0 (resp. t t 0 ) Let Ω be an open subset of R N and A an N-function. The Orlicz class K A (Ω) is defined as the set of real-valued measurable functions u on Ω such that, Ω A(u(x)) dx <.

3 On the regularity of solutions to the Poisson equation in Orlicz-spaces 3 The Orlicz-space L A (Ω) is the set of (equivalence classes of) real valued measurable functions u such that u K A(Ω), for some = (u) > 0. It s a Banach space under the norm u A,Ω = inf{ >0: A( u(x) )dx }. Ω The closure in L A (Ω) of the bounded measurable functions with compact support in Ω is denoted by E A (Ω). The equality E A (Ω)=L A (Ω) holds if and only if A satisfies the 2 condition, for all t or for t large according to whether Ω has a finite measure or not. The dual space of E A (Ω) can be identified with L Ā (Ω) by means of the pairing Ω u(x)v(x)dx and the dual norm on LĀ (Ω) is equivalent to u Ā,Ω. We recall the Hlder s inequality u(x)v(x) dx 2 u A,Ω v A,Ω, Ω for all u L A (Ω) and all v L A (Ω). We say that u n converges to u in modular sense (denoted by u n u (mod)) in L A (Ω), if for some constant >0, A( u n u ) dx 0, asn +. Ω Lemma 2.. (see [6])- Let u be an element of L A (R N ) with 2u K A (R N ) and T y u the translation of u,.i.e.,t y u(x) =u(x y). Then: A(T y u u)dx 0, as y 0. R N 2.3- We define the Newtonian function P N (x) onir N \{0} by: x if N =; 2 log( x ) P N (x) =P N ( x ) = if N =2; 2π k N if N 3, x k N =(2 N)σ N,whereσ N is the measure of unit sphere of IR N. The function P N (x) which is locally integrable, is the elementary solution of the Poisson equation P N = δ on IR N, where δ is the Dirac measure. It s easy to prove that: and that: for N =,P (x) and dp (x) dx P N (x) = for N =2,P 2 (x) L p loc (IR2 ) (resp. P 2(x) x i, ( i N) σ N x N lie in L p loc (IR), for p ; L p loc (IR2 )), for all p< (resp. p<2); for N 3, P N (x) L p loc (IRN ) (resp P N(x) L p loc (IRN )), for all p< N p< N N ). (resp. Definition 2.2. (see.[4])- Let f be a distribution on IR N with compact support. The distribution P.N(f) =P N f is called the Newtonian potential of f. In particular, P N is the Newtonian potential of δ and each distribution u with compact support is the Newtonian potential of its laplacian.

4 4 E. Azroul A. Benkirane M. Tienari 3 Regularity of the solution and its first derivatives The following lemmas will be used below. The first can be found in [], we give its proof for more convenience. Lemma 3.. Let Ω be an open subset of IR N and let A be an N-function. If u n u(mod) in L A (Ω), thenu n.f u.f in L (Ω), for all f L A (Ω). Proof- Letf L A (Ω), i.e. there exists a constant > 0 such that A( f ) L (Ω). Also, there exists a constant 2 > 0 such that M( 2(u n u) ) 0 in L (Ω), 2 which implies that there exists a subsequence (u nk )of(u n ) satisfying: u nk ua.einω and A( 2(u n k u) ) h (x) a.e in Ω, 2 for some function h (x) L (Ω). On the other hand, by convexity of A we get A( u n k ) 2 A(2(u n k u) )+ 2 A(2u ). For > 2 such that A( 2u ) L (Ω), we have u nk (x) f(x) 2 A (h(x)) f(x) 2 [h(x)+a( f(x) )], 2 where h(x) = [h 2 (x)+a( 2u)] L (Ω). By virtue of Lebesgue theorem we get u nk.f u.f in L (Ω). Finally, we conclude for the original sequence (u n ) by a standard contradiction argument. Lemma 3.2. Let A be an N-function and let g L loc A (IR N ). )- If f is a Radon measure on IR N with compact support, then the convolution g f is a function which lies in L loc A (IRN ). 2)- If f L A (IR N ) with compact support, then g f is continuous on IR N.

5 On the regularity of solutions to the Poisson equation in Orlicz-spaces 5 Proof- Let R>0 such that supp(f). Using the Lebesgue decomposition theorem we can write, f = v.m, where v is a function taking the values or - and m is the variation of f. Givingarealr>0 and a continuous function ξ on IR N with a support in B(0,r), we have g f,ξ B(0,r) ( where u, v = u(x)v(x) dx. ( ( B(y,r) g(x y)v(y) dm(y)) ξ(x) dx = g(x). ξ(x) dx) dm(y) dm) g A,B(0,R+r). ξ A,B(0,r), For 2), there exists a real > 0 such that g K A(). Moreover, g 2 f(x ) g 2 f(x 2) T x2 x ( g 2 )(z) g 2 (z). f(x 2 z) dz. Since, T x2 x ( g 2 ) g 2 0(mod) in L A(), asx 2 x, (due to lemma 2.) then the conclusion follows directly from lemma 3.. Definition 3.3. Let A be an N-function. We say that A satisfies the property (P) (resp. (P2)) if + A(t) dt < (resp. + t + N A(t) t + N N dt < ). Remarks 3.4. Note that: (P2) (P). Lemma 3.5. Let A be an N-function and α>. The following conditions are equivalent A(s) ds < (3.) s+α

6 6 E. Azroul A. Benkirane M. Tienari s ( Ā(s) )α ds <. (3.2) Proof. For the first implication, we have which implies that which is equivalent to and also to 2c lim c c A(s) ds =0, s+α A(c) lim =0, c c α lim c lim c c A (c) α =0 c α =0, Ā(c) α where we have used the following inequalities t A (t)ā (t) 2t. (3.3) The change of the integration variable and the integration by parts allow to have A(s) s ds = t +α (A (t)) +α da (t) = c 2 + dt α c (A (t)), α for some constants c and c 2. Hence by (3.3), we can write A(s) s +α ds α c c (Ā (t) 2t ) α dt = α2 α c 3 ( s )αdā(s) Ā(s) s = c 4 + c 5 ( c 3 Ā(s) )α ds, for some constants c 3, c 4 and c 5. Then, (3.) implies (3.2). For the reverse implication, doing the above steps in reverse order we can deduce analogously. Inthefollowing,wedenotebyord(f) the order of the distribution f. Theorem 3.6. Let Ω be an open subset of IR N (N 3), f be a distribution on Ω and u the solution of the equation (E). Then )- If ord(f)=0,thenu L loc u A (Ω) (resp. L loc A (Ω)) for all N-function A satisfying (P) (resp. all N-function A satisfying (P2)); 2)- if ord(f)=, then u L loc A (Ω) for all N-function A satisfying (P2); 3)- if f L loc A (Ω) and ( t ) 2 A(t) dt <, thenu is continuous; 4)- if f L loc A (Ω) and ( t ) N A(t) dt <, thenu is continuously differentiable.

7 On the regularity of solutions to the Poisson equation in Orlicz-spaces 7 Proof. Given an open bounded U such that U Ω and a function ϕ D(Ω) such that ϕ onu, we consider the functions F 0 = ϕ.f and F = (ϕ.u) F 0. Let u 0 and u be the Newtonian potentials of F 0 and F respectively. We have u 0 = P N F 0 = P N (ϕ.f) andu = P N F = P N ( (ϕ.u)) P N F 0. This implies that u 0 + u = P N ( (ϕ.u)) = u on U. Moreover, u is harmonic on U (because F =0onU), and so u and u 0 have similar regularity (locally). Hence, one can assume that Ω = IR N, f is a distribution with compact support and u = P.N(f). Note that if ord(f), i.e. N f k f = f 0 +, x k= k where f k (0 k N) are measures with compact support, then we have N u = u k, k=0 where u 0 = P.N(f 0 )andu k = P.N( f k x k ). ) Suppose that ord(f)=0, i.e. f = f 0. If A satisfies (P), then we claim that Indeed, for all real R>0wehave x <R P N L loc A (IRN ). R A( k N.P N (x) ) dx = σ N r N A( ) dr 0 r σ + N A(t) = dt <, N 2 R t + N. This implies by using lemma 3.2 that u = P N f L loc A (IR N ). Similarly, if A satisfies (P2), then P N (x) L loc A (IR N ), (for i N). Indeed, for all real R>0wehave P N (x) A(σ N ) dx x <R A( ) dx x <R x N R = σ N r N A( ) dr 0 rn σ + N A(t) dt <, N R N t + N N

8 8 E. Azroul A. Benkirane M. Tienari then, the conclusion follows from lemma ) Assume that ord(f)= and A satisfies (P2), then P N (x) L loc A (R N )for i N and (since (P2) implies (P)) P N L loc A (R N ), hence u L loc A (R N ). For3)and4)Takingα = N (resp. α = N ) in lemma 3.5, one deduce that N Ā satisfies (P ) (resp. (P 2 )), which implies as above that P N L loc A (RN ) (resp. P N L loc A (RN )). Then by using the lemma 3.2 for f and g = P N (resp. f and g = P N ), we conclude. Remark 3.7. In the L p case, the property (P) (resp. (P2)) is equivalent to (resp. p< N p< N N ). Example 3.8. We consider the N-functions: t N A(t) = (log( + t)) and B(t) = t α (log( + t)), α where α >. The N-function A (resp. B) satisfies the property (P) (resp. (P2)). Indeed, N N and A(t) t + N B(t) t + N N ) dt = ) dt = t(log( + t)) α dt <, t(log( + t)) α dt <. Theorem 3.9. Let Ω be an open subset of IR 2,andA be an N-function such that + A(t) dt <, e2t for some >0. If f is a measure on Ω (resp. L loc (Ω)), then every solution u of A the equation (E) lies in L loc A (Ω) (resp. is continuous on Ω). Proof - As in the proof of theorem 3.6, we can assume that Ω = IR 2, supp(f) is compact and u = P.N(f). First, we have P 2 L loc A (IR 2 ). Indeed, for all real R>0wehave x <R A( 2π P 2(x)) dx R = σ 2 ra( log(r) 0 ) dr + = σ 2 r A(log(r) 3 ) dr = σ 2 R + c A(t) dt <, e2t where c = log(r), then we conclude by lemma 3.2.

9 On the regularity of solutions to the Poisson equation in Orlicz-spaces 9 Example 3.0. We consider the N-function A(t) =e t t, its conjuguate is A(t) =(+ t )log(+ t ) t. The N-function A satisfies the condition of theorem 3.9. Indeed, for R>0and> 2 + we have A(t) + e dt 2t dt <. e (2 )t 4 Maximal regularity of radial solutions Inthefollowing,wenotebyB(0,R 0 )anopenballofr N with radius R 0 > 0which is equal to R N for R 0 =+. Theorem 4.. Let A be an N-function and f L A (B(0,R 0 )). radial solution u of the equation Then every u = f inb(0,r 0 ), (4.) satisfies L loc A (B(0,R 0 ) {0}). If in addition A( f(x) )log( x ) is integrable then, L loc A (B(0,R 0 )). (4.2) The proof uses the following lemma (see. [4]). Lemma 4.2. Let f be a radial and integrable function on R N with compact support. The Newtonian potential P.N(f) is continuously differentiable on R N {0} and P.N(f)(x) =P N (x) f(y)dy + P N (y)f(y)dy B(0, x ) R N B(0, x ) x P.N(f)(x) = f(y)dy. σ N x N B(0, x ) The second derivatives of P.N(f) are locally integrable on R N {0} and 2 P.N(f) (x) = x ix j x f(x)+(δ ij 2 N x ix j x ) N f(y)dy. 2 σ N x N B(0, x ) Proof of Theorem 4.. LetB be an open ball such that B B(0,R 0 )andlet Φ be a radial smooth function with compact support in B(0,R 0 ) such that Φ in B. We consider the functionf 0 =Φ.f and f = (Φu) f 0.Wehave P.N(f 0 )+P.N(f )=P N ( (Φu)) = u on B.

10 0 E. Azroul A. Benkirane M. Tienari Since P.N(f )isharmoniconb,thenp.n(f 0 )andu have the same regularity on B. Hence, we can assume that f is radial with compact support and u = P.N(f). Let 0 <R<R 0 and let ε>0. We shall show that L A (B(ε, R)), (4.3) where B(ε, R) ={x R N : ε< x < R}. It follows directly from lemma 4.2 that, f(x) + C f(y) dy, (4.4) x N B(0, x ) for some positive constant C. Hence, by virtue of convexity of A we have where and B(ε,R) I 2 = A( 2 I = B(ε,R) )dx 2 I + 2 I 2, B(ε,R) A( C x N A( f(x) )dx B(0, x ) f(y) dy)dx. Since I < it s sufficient to show that I 2 <. For that we have by Jensen s inequality and Fubini s theorem, I 2 B(ε,R) ( C 2 x N = (A( f(y) = C 3 A( f(y) B(0, x ) ) max(ε, y )< x <R A( f(y) )dy)dx C 2 x N dx)dy R )log( max(ε, y ) )dy, for some positive constants C 2 and C 3, which implies (4.3). For the second part of theorem, assumming now that A( f(x) )log( x ) is integrable. As above by Jensen s inequality and Fubini s theorem we get for 0 <R<R 0, 2 = 2 = 2 A( 2 )dx A( f(x) )dx + ( C 4 2 x N A( f(x) )dx + (A( f(y) 2 B(0, x ) ) A( f(y) )dy)dx y < x <R C 4 x N dx)dy A( f(x) )dx + C 5 A( f(y) )log( R y )dy,

11 On the regularity of solutions to the Poisson equation in Orlicz-spaces for some positive constants C 4 and C 5, which achieves the proof of theorem 4.. Theorem 4.3. Let A be an N-function. If f L A (B(0,R 0 )) (with R 0 < ), then every radial solution u of the equation (4.) satisfies: L A (B(ε, R 0 )), for all ε>0. If in addition A( f(x) )log( x ) is integrable, then L A (B(0,R 0 )). Proof. Replacing R by R 0 in the proof of theorem 4., we obtain the desired statement. Theorem 4.4. Let A be an N-function such that A satisfies the 2 condition and let f L A (B(0,R 0 )). Then every radial solution u of the Poisson equation (4.) satisfies L A (B(0,R 0 )). Proof. Using lemma 4.2, we have f(x) + C f(y) dy x N B(0, x ) = f(x) +C Tf(x), where the mapping T is defined by the formula Tf(x) = x N B(0, x ) f(y) dy. Since T is of weak type (,) and of type (, ), then the interpolation theorem III. of [2] implies that Tf L A (B(0,R 0 )),which gives the conclusion. Remark 4.5. The previous argument cannot be generalized for arbitrary N- function (compare Remark 5. p.290 in [4]). In fact, let B = B(0, ) 2 R3 and consider the N-function A(t) =(+t)log(+t) t. It s easy to verify that the function f(x) = L x 3 (log x ) 2 A (B), but Tf(x) = x 3 B(0, x ) f(y) dy is not in L A (B).

12 2 E. Azroul A. Benkirane M. Tienari References [] R.Adams, Sobolev spaces, Acad. press (975). [2] A.Benkirane, Potentiel de Riesz et Problmes elliptiques dans les espaces d Orlicz, Thse de Doctorat, Universit Libre de Bruxelles, 988. [3] A.Benkirane and J.P.Gossez, An approximation theorem in higher Orlicz- Sobolev spaces and application, Studia Math. 92 (989) pp [4] R.Dautray et J.L.Lions, Analyse Mathmatique et calcul numrique, Volume 2, Masson (987). [5] D.Gilbarg and N.S.Trudinger, Elliptic partial differential equations of second ordre, Springer Verlag (983). [6] J.P.Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math. 74 (982) pp [7] J.P.Gossez, Non linear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 90 (974), pp [8] Grter and Wildman, Manuscripa Math 37 (982) [9] M.Krasnosel skii and Ya.Rutickii, Convex functions and Orlicz spaces, Noordhoff, 96. [0] A.Kufner, O.John and S.fucik, Function spaces, Academia, 977. [] M.Tienari, A degree theory for a class of mappings of monotone type in Orlicz- Sobolev spaces, Ann. Acad. Scientiarum Fennice Helsinki (994). E. Azroul and A. Benkirane Dpartement de Mathmatiques et Informatique, Facult des Sciences Dhar Mahraz B.P.796 Atlas, Fs, Maroc M. Tienari Department of Mathematical Sciences University of Oulu, FIN Oulu, Finland.