×

A Lipschitzian estimate at the boundary points for solutions of quasilinear equations of divergent form. (English. Russian original) Zbl 0677.35035

Sib. Math. J. 28, No. 4, 632-639 (1987); translation from Sib. Mat. Zh. 28, No. 4(164), 145-153 (1987).
In a bounded domain \(\Omega \subset {\mathbb{R}}^ n\) the authors study the problem \[ (1)\quad da_ i(x,u,u_ x)/dx_ i+a(x,u,u_ x)=0\quad (x\in \Omega),\quad u(x)=\phi (x)\quad x\in \partial \Omega. \] Under weak enough conditions the authors prove the estimate \(| u(x)-\phi (x_ 0)| \leq c| x-x_ 0|\) \((x_ 0\in \partial \Omega\), \(x\in \Omega)\) for a bounded generalized solution of the problem (1). It is supposed that \[ \nu (1+| p|)^{m-2}\xi^ 2\leq \partial a_ i(x,u,p)/\partial p_ j\xi_ i\xi_ j\leq \mu (1+| p|)^{m- 2}\xi^ 2,\quad | a_ i(x,u(x),0)| \leq \mu_ 1, \]
\[ | a_ i(x,u(x),p)-a_ i(y,u(y),p)| \leq \mu_ 1| x- y|^{\alpha}(1+| p|)^{m-2}, \]
\[ | a(x,u(x),p)| \leq \mu_ 2| p|^ m+\Phi (x)(1+| p|)^{m/2}, \] \(\Phi\in L_ q(\Omega)\), \(m\geq 2\), \(\nu >0\), \(\mu >0\), \(\mu_ 1\geq 0\), \(\mu_ 2\geq 0\), \(\alpha\in (0,1)\), \(q>n\) and that either \(\Omega\) is convex and \(\phi =0\) or \(\partial \Omega \in C^{1+\beta}\), \(\phi \in C^{1+\beta}(\partial \Omega)\). The constant \(c>0\) depends only on ess sup\(| u|\), \(\nu\), \(\mu\), \(\mu_ 1\), \(\mu_ 2\), m, \(\| \Phi \|_{q,\Omega}\), \(\delta =1-n/q\), \(\beta\), \(\| \phi \|_{C^{1+\beta}(\partial \Omega)}\), \(\partial \Omega\). A similar result is stated also for the Dirichlet problem for a parabolic equation \[ -u_ t+da_ i(x,t,u,u_ x)/dx_ i+a(x,t,u,u_ x)=0 \] in a cylinder \(\Omega\) \(\times (0,T)\).
Reviewer: J.Rojtberg

MSC:

35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35J15 Second-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35K10 Second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] O. A. Ladyzhenskaya and N. N. Ural’tseva, Estimates on the Boundary of a Domain for the Hölder Norms of the Derivatives of Solutions of Quasilinear Elliptic and Parabolic Equations of General Form [in Russian], No. P-1-85, Preprint, Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, Leningrad (1985).
[2] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press (1968).
[3] M. Giaquinta and E. Giusti, ?Global C1+?-regularity for second-order quasilinear elliptic equations?, J. Reine Angew. Math., No. 351, 55-65 (1984). · Zbl 0528.35014
[4] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. (1968).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.