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Stable Solutions of Semilinear Elliptic Problems in Convex Domains
| Content Provider | Semantic Scholar |
|---|---|
| Author | Abstract, Sagun Chanillo |
| Abstract | In this note we consider semilinear equations ?u = f (u), with zero Dirichlet boundary condition, for smooth and nonnegative f , in smooth, bounded, strictly convex domains of R N. We study positive classical solutions that are semi-stable. A solution u is said to be semi-stable if the linearized operator at u is nonneg-ative deenite. We show that in dimension two, any positive semi-stable solution has a unique, nondegenerate, critical point. This point is necessarily the maximum of u. As a consequence, all level curves of u are simple, smooth and closed. Moreover, the nondegeneracy of the critical point implies that the level curves are strictly convex in a neighborhood of the maximum of u. Some extensions of this result to higher dimensions are also discussed. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www-ma1.upc.es/~cabre/chanillo.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Arabic numeral 0 Commutation theorem Convex function Critical point (network science) Dimensions Navier–Stokes equations Semiconductor industry Semilinear response Solutions |
| Content Type | Text |
| Resource Type | Article |
