Bounded weak solutions to nonlinear elliptic equations.
The local boundedness of weak solutions to variational inequalities (obstacle problem) with the linear growth condition is obtained. Consequently, an analogue of a theorem by Reshetnyak about a.eḋifferentiability of weak solutions to elliptic divergence type differential equations is proved for variational inequalities.
In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated -Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.
We study connections between Sobolev space solutions of the Dirichlet problem for semilinear second order elliptic equations in divergence form and solutions of backward stochastic differential equations with random terminal time.
The role of the second critical exponent , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem , under zero Dirichlet boundary conditions, in a domain in with bounded, smooth boundary. Given , a geodesic of the boundary with negative inner normal curvature we find that for , there exists a solution such that converges weakly to a Dirac measure on as , provided that is nondegenerate in the sense of second variations of...
We prove the Hölder continuity of the homogeneous gradient of the weak solutions of the p-Laplacian on the Heisenberg group , for .
We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures for which the Sobolev space can be imbedded into . The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the theory of singular...
We show the equivalence of some different definitions of p-superharmonic functions given in the literature. We also provide several other characterizations of p-superharmonicity. This is done in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. There are many examples of such spaces. A new one given here is the union of a line (with the one-dimensional Lebesgue measure) and a triangle (with a two-dimensional weighted Lebesgue measure). Our results also...