Loading...
Please wait, while we are loading the content...
Similar Documents
Two-dimensional Nonlinear Boundary Value Problems for Elliptic Equations
| Content Provider | Semantic Scholar |
|---|---|
| Author | Lieberman, Gary M. |
| Copyright Year | 2010 |
| Abstract | Boundary regularity of solutions of the fully nonlinear boundary value problem F(x,u,Du, D2u) = 0 inn, G(x,u, Du) = 0 on dO is discussed for two-dimensional domains Q. The function F is assumed uniformly elliptic and G is assumed to depend (in a nonvacuous manner) on Du. Continuity estimates are proved for first and second derivatives of u under weak hypotheses for smoothness of F, G, and 0. In [9] nonlinear boundary value problems for nonlinear, uniformly elliptic equations were studied, and several important existence and regularity results were proved when the boundary condition is oblique, i.e., it prescribes a nontangential directional derivative. Results were derived there for problems in any number of dimensions, but it was shown that the two-dimensional case is simpler than the higher-dimensional one. Here we examine the two-dimensional case in more detail using different arguments. By exploiting special features of the two-dimensional problem, we can weaken the regularity hypotheses in [9] and, more significantly, remove the obliqueness assumption. We refer the reader to [1 and 14] for existence results with nonoblique boundary condition; our main concern here is with the regularity of solutions. Specifically we consider the problem (1) F[u] = F(x,u,Du,D2u) =0 in fi, G(x,u,Du) = Q on dfi for a bounded open subset fi of R2. Leaving aside temporarily questions of smoothness of F, G, and fi, we assume that there are positive constants p and x, and a positive function A for which (2) X(x,z,p)I X for all (x, z,p, r) € fi x R2 x S2 and (x',z,p) e dfi x R x R", where S2 is the set of all real symmetric 2x2 matrices and subscripts denote partial derivatives. (Actually these conditions can be generalized slightly at the cost of additional technical complications; see [9, §§2 and 4].) Received by the editors February 21, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 35J60, 35J65; Secondary 35J67, 35B45. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ams.org/journals/tran/1987-300-01/S0002-9947-1987-0871676-8/S0002-9947-1987-0871676-8.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Assumed Belief revision Courant–Friedrichs–Lewy condition Dimensions Directional derivative Emoticon Estimated Mathematics Subject Classification Neoplasm Metastasis Nonlinear system Oblique projection Scott continuity Solutions Subgroup |
| Content Type | Text |
| Resource Type | Article |
