Hölder regularity for the general parabolic p(x, t)-Laplacian equations
Tóm tắt
In this paper we obtain the local Hölder regularity of the gradient of weak solutions for the general parabolic p(x, t)-Laplacian equations
$$u_t-\text{div}\ \mathcal{A}\left( \nabla u, x, t \right)\,=\,\text{div} \left(|\mathrm{\bf f}|^{p(x, t)-2} \mathrm{\bf f}\right),$$
provided p(x, t),
$${\mathcal{A}}$$
and
$${\mathrm{\bf f}}$$
satisfy some proper conditions. More precisely, we shall prove that
$$\nabla u \in C_{loc}^{0;\alpha,\alpha/2}(\Omega_T)\,\mbox{for some} \, \, \alpha \in (0, 1). $$
Tài liệu tham khảo
Acerbi E., Mingione G.: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)
Acerbi E., Mingione G.: Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math 584, 117–148 (2005)
Acerbi E., Mingione G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136, 285–320 (2007)
Bögelein V., Duzaar F.: Higher integrability for parabolic systems with non-standard growth and degenerate diffusions. Publ. Mat. 55(1), 201–250 (2011)
Bögelein, V., Duzaar, F.: Hölder estimates for parabolic p(x, t)-Laplacian systems. Math. Ann. (to appear)
Bögelein, V., Duzaar, F., Mingione, G.: The regularity of general parabolic systems with degenerate diffusion. Mem. Am. Math. Soc. 221(1041), vi+143 (2013)
Byun S., Wang L.: Quasilinear elliptic equations with BMO coefficients in Lipschitz domains. Trans. Am. Math. Soc. 359(12), 5899–5913 (2007)
Byun S., Wang L, Zhou S: Nonlinear elliptic equations with BMO coefficients in Reifenberg domains. J. Funct. Anal. 250(1), 167–196 (2007)
Challal S., Lyaghfouri A.: Second order regularity for the p(x)-Laplace operator. Math. Nachr. 284(10), 1270–1279 (2011)
Challal S., Lyaghfouri A.: Gradient estimates for p(x)-harmonic functions. Manuscripta Math. 131(3–4), 403–414 (2010)
Coscia A., Mingione G.: Hölder continuity of the gradient of p(x)-harmonic mappings. C. R. Acad. Sci. Paris Math. 328(4), 363–368 (1999)
Diening L.: Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces L p(.) and W k,p(.). Math. Nach. 268(1), 31–43 (2004)
Diening L., Ru̇žička M.: Calderón-Zygmund operators on generalized Lebesgue spaces L p(·) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003)
Diening L., Ru̇žička M.: Integral operators on the halfspace in generalized Lebesgue spaces L p(.), part I. J. Math. Anal. Appl. 298(2), 559–571 (2004)
Duzaar F., Mingione G.: Gradient estimates via non-linear potentials. Am. J. Math. 133(4), 1093–1149 (2011)
Fan X., Shen J., Zhao D.: Sobolev embedding theorems for spaces W k,p(x)(Ω). J. Math. Anal. Appl. 262, 749–760 (2001)
Fan X., Zhao D.: On the spaces L p(x)(Ω) and W m,p(x)(Ω). J. Math. Anal. Appl. 263, 424–446 (2001)
Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press, Princeton (1983)
Harjulehto P.: Variable exponent Sobolev spaces with zero boundary values. Math. Bohem. 132, 125–136 (2007)
Kinnunen J., Lewis J.L.: Higher integrability for parabolic systems of p-Laplacian type. Duke Math. J. 102(2), 253–271 (2000)
Kinnunen J., Zhou S.: A local estimate for nonlinear equations with discontinuous coefficients. Comm. Partial Differ. Equ. 24, 2043–2068 (1999)
Lyaghfouri A.: Hölder continuity of p(x)-superharmonic functions. Nonlinear Anal. 73(8), 2433–2444 (2010)
Misawa, M: Local Höder regularity of gradients for evolutional p-Laplacian systems. Ann. Mat. Pura Appl. (4) 181(4), 389–405 (2002)
Misawa M.: L q estimates of gradients for evolutional p-Laplacian systems. J. Differ. Equ. 219(2), 390–420 (2005)
Mingione G.: Gradient estimates below the duality exponent. Math. Ann. 346(3), 571–627 (2010)
Palagachev D.: Quasilinear elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 347, 2481–2493 (1995)
Phuc, N.C.: Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10(1), 1–17 (2011)
Rajagopal K.R., Růžička M.: Mathematical modeling of electro-rheological materials. Contin. Mech. Thermodyn 13(1), 59–78 (2001)
Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748. Springer, Berlin, (2000)
Xu M., Chen Y.: Hölder continuity of weak solutions for parabolic equations with nonstandard growth conditions. Acta Math. Sin. (Engl. Ser.) 22(3), 793–806 (2006)
Yao F.: Local Hölder regularity of the gradients for the elliptic p(x)-Laplacian equation. Nonlinear Anal. 78, 79–85 (2013)
Zhang C., Zhou S.: Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian. J. Math. Anal. Appl. 389(2), 1066–1077 (2012)