1,721,731 research outputs found
Boundary point lemmas and overdetermined problems
No full textAbstractTwo elliptic boundary value problems are considered: a problem of mixed type in a cylindrical domain, and a Dirichlet problem in an annular domain. Under some overdetermined conditions on the boundary gradient, symmetry results for domain and solution are proved. The method of proof involves the classical boundary point lemma by Hopf, as well as a suitable adaptation of it that works well at certain corners
Review of "Rearrangements and applications to symmetry problems in PDE", a paper by F. Brock
No full textReview of "Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry", a paper by Naoki Shioji and Kohtaro Watanabe
No full textReview of "On symmetry properties of parabolic equations in bounded domains", a paper by Juraj Földes
No full textReview of "Steady states and standing pulses of a skew-gradient system", a paper by Ya-Ping Lin and Shyuh-yaur Tzeng
No full textConvex functions over the whole space locally satisfying fractional equations
No full textWe investigate the structure of convex functions over the whole space which satisfy in some convex domain an equation involving the fractional Laplacian. Roughly speaking, it turns out that such solutions are either strictly convex in the given domain, or degenerate in the sense that their graph is a ruled hypersurface. We also consider regular solutions, that some fractional equations admit, and show that the convexity of the datum is transmitted to the solution through its regularity. The results are obtained by means of a fractional form of the celebrated \textit{convexity maximum principle} devised by Korevaar in the 80's. More precisely, we construct an anisotropic, degenerate, fractional operator that nevertheless satisfies a maximum principle, and we apply such an operator to the concavity function associated to the solution. An explicit, two-dimensional example is also constructed
Review of "Hölder stability for Serrin’s overdetermined problem", a paper by Giulio Ciraolo, Rolando Magnanini and Vincenzo Vespri
No full textReview of "Overdetermined boundary value problems for the ∞-Laplacian", a paper by Giuseppe Buttazzo and Bernd Kawohl
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