POINCARE INEQUALITY AND CAMPANATO ESTIMATES FOR WEAK SOLUTIONS OF PARABOLIC EQUATIONS

Size: px

Start display at page:

Download "POINCARE INEQUALITY AND CAMPANATO ESTIMATES FOR WEAK SOLUTIONS OF PARABOLIC EQUATIONS"

Jasmine Allison
5 years ago
Views:

1 Electronic Journal of Differential Equation, Vol. 206 (206), No. 204, pp. 8. ISSN: URL: or POINCARE INEQUALITY AND CAMPANATO ESTIMATES FOR WEAK SOLUTIONS OF PARABOLIC EQUATIONS JUNICHI ARAMAKI Abtract. We hall how that the Poincaré type inequality hold for the weak olution of a parabolic equation. The key i to control the L p norm of the firt derivative of the weak olution with repect to the time variable. The inequality i neceary to get an etimate in the Campanato pace L p,µ for general parabolic equation.. Introduction Let B r (x 0 ) R n be a ball with center x 0 and the radiu r > 0. Let u W,p (B r (x 0 )) and define u x0,r udx. B r (x 0 ) B r(x 0) Then we have the well known Poincaré inequality u u x0,r p dx Cr p B r(x 0) B r(x 0) u p dx, (.) where the contant C depend on n and p, but i independent of r and u (ee, for example, Chen and Wu [, Appendix I Corollary 3.]). Thi type etimate i ued in the Campanato pace approach to obtain the Hölder regularity of weak olution for elliptic or quailinear elliptic ytem with degeneracie. For example, for the elliptic ytem in divergence form, ee Giaquinta [7], and the regularity of olution for the quailinear elliptic ytem with degeneracy a p-laplacian ha been treated recently in Giacomoni et al. [4, 5, 6] who eentially ued the technique of the Campanato etimate of the Lieberman [9, p. 2] and [7, p. 45], uing the Poincaré inequality (.). See, for example, [6, Theorem A. in the Appendix]. We are intereted in the Hölder regularity of weak olution for parabolic equation. To apply the Campanato etimate in thi cae, it uffice to ue the following Poincaré type inequality. To explain preciely, let z 0 (x 0, t 0 ) Q T Ω (0, T ). If we put u z0,r u dx dt Q r (z 0 ) 200 Mathematic Subject Claification. 35A09, 35K0, 35D35. Key word and phrae. Poincaré type inequality; weak olution; parabolic equation. c 206 Texa State Univerity. Submitted December 24, 205. Publihed July 28, 206.

2 2 J. ARAMAKI EJDE-206/204 for any cylinder Q r (z 0 ) B r (x 0 ) (t 0, t 0 + r 2 ] Q T, the following inequality u u z0,r p dx dt Cr p u p dx dt, (.2) where u i the gradient of u with repect to the pace variable x doe not hold for a general function u(x, t) L p (0, T ; W,p (Q r (z 0 )). Thi i the fundamental difference from the elliptic theory. However, when u i a weak olution of a parabolic equation, by uing the equation and combining the Poincaré inequality (.) with repect to the pace variable, we hall how that the inequality (.2) hold. Such inequality i ued in the Campanato pace L p,µ etimate for weak olution of a parabolic equation. In fact, Yin [] ued the inequality for p 2 and conducted u to L 2,µ -etimate for weak olution of parabolic equation. We are convinced that we can ue the general inequality (.2) for L p.µ -etimate for parabolic equation. It will appear in the future work. 2. Main reult In thi ection, we give the main theorem of thi paper. Let Ω be a bounded domain in R n with C boundary Ω, and define Q T Ω (0, T ) with T > 0. We conider the parabolic equation n ( ) u t aij (x, t)u xj x i 0 in Q T (2.) i,j where a ij L (Q T ) atifie the ellipticity condition: there exit contant a 0 and A 0 with 0 < a 0 A 0 < uch that n a 0 ξ 2 a ij (x, t)ξ i ξ j A 0 ξ 2 i,j for all (x, t) Q T and ξ R n. We hall conider the weak olution of (2.). Definition 2.. Let p <. We ay u L p (0, T ; W,p (Ω)) i a weak olution of (2.) if the following equality hold. ( n ) uv t + a ij u xj v xi dx dt 0 (2.2) Q T i,j for all v W,p (0, T ; W,p 0 (Ω)) with v(x, 0) v(x, T ) 0 where p i the conjugate exponent of p, i.e., p + p. Here W,p (Ω) and W,p 0 (Ω) denote the tandard Sobolev pace. We note that ince W,p (0, T ; W,p 0 (Ω)) C 0 ([0, T ]; W,p 0 (Ω)) according to the Sobolev embedding theorem, the value v(x, 0) and v(x, T ) are meaningful. We ue ome tandard notation. We will denote any point in Q T by z (x, t), and for z (x, t ), z 2 (x 2, t 2 ) Q T, the parabolic ditance i defined by dit(z, z 2 ) max{ x x 2, t t 2 /2 }. For r > 0 and z 0 (x 0, t 0 ) Q T, we write B r (x 0 ) {x R n ; x x 0 < r}, Q r (z 0 ) B r (x 0 ) (t 0, t 0 + r 2 ).

3 EJDE-206/204 POINCARÉ TYPE INEQUALITY 3 We alo write the average of a function u on Q r (z 0 ) by u z0,r u dx dt, Q r (z 0 ) where Q r (z 0 ) r 2 B r (x 0 ) and B r (x 0 ) i the volume of B r (x 0 ). We are in a poition to tate the main theorem. Theorem 2.2. Let < p <, u L p (0, T ; W,p (Ω)) be a weak olution of (2.), and z 0 (x 0, t 0 ) Q T with Q 2r (z 0 ) Q T. Then there exit a contant C > independent of r and u uch that u u z0,r p dz Cr p u p dz, Q 2r(z 0) where u denote the gradient of u with repect to the pace variable x. 3. Proof of Theorem 2.2 In thi ection, we ue the technique baed on [0, Lemma 3 and 4]. We aume that u L p (0, T ; W,p (Ω)) i a weak olution of (2.) and z 0 (x 0, t 0 ) Q T and Q 2r (z 0 ) Q T. By a tranlation, we aume that z 0 (0, 0), and we write Q r Q r (0, 0) and B r B r (0) for the brevity of notation. Chooe a mooth cut-off function σ(x) uch that { if x r, σ(x) (3.) 0 if x 2r, 0 σ(x), σ(x) 2/r, and σ(x) σ( x ) i monotone decreaing with repect to x. For 0 < t < r 2, let χ [,t] (τ) be the characteritic function of [, t] and define u σ Q r 2r uσdz Q 2r σdz Q 2r uσdz (2r) 2 σ dx, and where Q r (t) {(x, t); x < r}. Define where u σ Q r,t 2r(t) uσ dx Q σ dx, 2r(t) φ(x, τ) σ(x)χ [,t] (τ) ign(u σ r,t u σ r,) u σ r,t u σ r, p, for a > 0, ign(a) 0 for a 0, for a < 0. Uing the Steklov averaging to approximate φ and then taking the limit, we can ue φ(x, τ) a a tet function of (2.2) (cf. [0] or []), o formally ( n ) uφ τ + a ij (x, τ)u xj φ xi dx dτ 0. (3.2) Q T i,j We ue the following two lemma. The firt i a weighted Poincaré inequality with repect to the pace variable.

4 4 J. ARAMAKI EJDE-206/204 Lemma 3.. Let the function σ be a in (3.), u(x) W,p ( ) for ome p < and define B u 2r,σ 2r uσ dx σ dx. Then we have u u 2r,σ p σ dx C(n, p)r p u p σ dx. For a proof of the above lemma, ee Lieberman [8, Lemma 6.2]. Lemma 3.2. Let p < and u L p (Q r (z 0 )). Then we have u u z0,r p dz 2 p u L p dz for any L R. Proof. By the triangle inequality, for any L R, we have ( ) /p u u z0,r p dz ( ) /p ( /p. u L p dz + u z0,r L dz) p By the definition of u z0,r and the Hölder inequality, u z0,r L p dz u z0,r L p Q r (z 0 ) Q r (z 0 ) (u L)dz p Q r (z 0 ) Q r (z 0 ) p[( u L p dz Q r(z 0) Q r (z 0 ) p+p/p u L p dz u L p dz. ) /p Qr (z 0 ) /p ] p Here we ued p + p/p 0. Thu we get the concluion. We hall etimate each term of (3.2). We have uφ τ dx dτ Q T u τ (x, τ)σ(x)χ [,t] (τ)ign(u σ r,t u σ r,) u σ r,t u σ r, p dxdτ Q T t u τ (x, τ)σ(x)dxdτ ign(u σ r,t u σ r,) u σ r,t u σ r, p [ Ω Q 2r(t) u(x, t)σ dx Q 2r() ign(u σ r,t u σ r,) u σ r,t u σ r, p ] u(x, )σ dx

5 EJDE-206/204 POINCARÉ TYPE INEQUALITY 5 Thu we have Ω Ω σ dx (u σ r,t u σ r,)ign(u σ r,t u σ r,) u σ r,t u σ r, p σ dx u σ r,t u σ r, p. uφ τ dx dτ σ(x)dx u σ r,t u σ r, p Q T Ω σ(x)dx u σ r,t u σ r, p B r c 0 r n u σ r,t u σ r, p for ome c 0 > 0. On the other hand, we put n I a ij (x, τ)u xj (x, τ)φ xi (x, τ)dxdτ. Q T i,j Since a ij L (Q T ), we have t I C u(x, τ) σ(x) u σ r,t u σ r, p dx dτ. (3.3) If we write σ(x) σ(x) /p σ(x) /p, and apply the Hölder inequality, we have ( t ) /p I u(x, τ) p σ(x) dx dτ ( t ) /p σ(x) u σ r,t u σ r, p dx dτ. Here we ue the Young inequality (cf. Evan [2, p. 706]): ab δa p + (δp ) p/p p b p for any a, b 0 and any δ > 0. Uing thi inequality, we have t I Cδ σ(x) u σ r,t u σ r, p dx dτ t + C(p ) p/p p δ p/p If we put ε rδ, uing t r 2, we have I 2Cεr 2 (t ) u σ r,t u σ r, p t + 2C(p ) p/p p ε p/p r +p/p t C εr n u σ r,t u σ r, p + C(p, ε)r p 2 u(x, τ) p σ(x) dx dτ. B2r B2r u(x, τ) p dx dτ u(x, τ) p dx dτ From (3.2), (3.3) and (3.4), if we chooe ε > 0 mall enough, we have t u σ r,t u σ r, p Cr n+p 2 (3.4) u(x, τ) p dx dτ. (3.5)

6 6 J. ARAMAKI EJDE-206/204 By Lemma 3.2 and an elementary inequality, we have u u z0,r p dx dt 2 p u u σ r p dx dt Q r Q r 4 p u u σ r,t p dx dt + 4 p Q r Q r u σ r,t u σ r p dx dt. Since σ 0 and σ(x) on B r, we have r 2 u u σ r,t p dx dt u u σ r,t p dx dt u u σ r,t p σ dx dt Q r 0 B r Q 2r According to Lemma 3., u(x, t) u σ r,t p dx B r u(x, t) Q 2r(t) u(y, t)σ(y)dy p σ(x)dx Q 2r(t) σ(y)dy C(n, p)r p u(x, t) p σ(x)dx. Hence we ee that u u σ r,t p dx dt Cr p Q r Since u σ Q r 2r uσdz (2r) 2 σ dx (2r) 2 0 B 2r u(x, )σ(x) dx d (2r) 2 σ dx (2r) 2 0 Q 2r() u(x, )σ(x) dx d (2r) 2 Q σ dx 2r() (2r) 2 (2r) 2 u σ r,d 0 u σ r, dy d, Q 2r Q 2r we have u σ r,t u σ r (u σ r,t u σ Q 2r r,) dy d. Q 2r Therefore, by the Hölder inequality, we have u σ r,t u σ r p Q 2r p (u Q σ r,t u σ r,) dy d 2r Q 2r p[( Q 2r Q 2r u(x, t) p dx dt. (3.6) p ) /p Q2r u σ r,t u σ r, p dy d /p ] p Q 2r p+p/p Q 2r u σ r,t u σ r, p dy d Q 2r Q 2r u σ r,t u σ r, p dy d.

7 EJDE-206/204 POINCARÉ TYPE INEQUALITY 7 Thu uing (3.5), we have u σ r,t u σ r p dx dt Q r Q 2r u σ r,t u σ r, p dy d dx dt Q r Q 2r t Q 2r r n+p 2 u(x, τ) p dx dτ dy d dx dt Q r Q 2r t Q 2r r n+p 2 u(x, τ) p dx dτ dy d dx dt Q r Q 2r t Q 2r Q r Q 2r r n+p 2 u(x, τ) p dx dτ Cr p Q 2r u(x, τ) p dx dτ. Here we ued the fact that Q r ω n r n+2 where ω n i the volume of the unit phere in R n. So we get u u z0,r p dx dt Cr p u(x, t) p dx dt. Q r Q 2r Thi complete the proof of Theorem 2.2. Acknowledgment. We would like to thank the anonymou referee for hi or her very kind advice about an early verion of thi article. Reference [] Y-Z. Chen and L-C. Wu, Second Order Elliptic Equation and Elliptic Sytem, AMS, Tranlation of Mathematical Monograph, Vol. 74, (99). [2] L. C. Evan; Partial Differential Equation, AMS, Graduate Studie in Mathematic, Vol. 9, (200). [3] L. C. Evan; Partial Differential Equation, AMS, Graduate Studie in Mathematic, Vol. 9, (200). [4] J. Giacomoni, I. Schindler, P. Taká c; Sobolev veru Hölder ocal minimizer and exitence of multiple olution for a ingular quailinear equation, Annali Scuola Norm. Sup. Pia, Ser. V, Vol. 6(), (2007), [5] J. Giacomoni, I. Schindler, P. Taká c; Régularité höldérienne pour de équation quailinéaire elliptique inguliére, Compte Rendu de l Académie de Science de Pari, Série I, Vol. 350, (2002), [6] J. Giacomoni, I. Schindler, P. Taká c; Singular quailinear elliptic ytem and Hölder regularity, Advance in Differential Equation, Vol. 20(3-4), (205), [7] M. Giaquinta; Introduction to Regularity Theory for Nonlinear Elliptic equation, Lecture note in Math. Birkhäuer, (993). [8] G. M. Lieberman; Second Order Parabolic Differential Equation, World Scientific, (2005). [9] G. M. Lieberman; Boundary regularity for olution of degenerate elliptic equation, Nonlinear Analyi, Theory, Method and Application, Vol. 2, No., (988), [0] M. Struwe; On the Hölder continuity of bounded weak olution of quailinear parabolic ytem, Manucripta Math. Vol. 5, (98), [] H. Yin; L 2,µ (Q)-etimate for parabolic equation and application, J. Partial Diff. Eq. Vol. 0, (997), 3-44.

8 8 J. ARAMAKI EJDE-206/204 Junichi Aramaki Diviion of Science, Faculty of Science and Engineering, Tokyo Denki Univerity, Hatoyama-machi, Saitama , Japan addre:

TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL

GLASNIK MATEMATIČKI Vol. 38583, 73 84 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian Haihen Lü, Donal O Regan and Ravi P. Agarwal Academy of Mathematic and Sytem Science, Beijing, China, National