On three-point boundary value problem with a weighted integral condition for a class of singular parabolic equations.
The paper is devoted to the estimate u(x,k)Kk{capp,w(F)pw(B(x,))} 1p-1, for a solution of a degenerate nonlinear elliptic equation in a domain , , , under the boundary-value conditions for , for and where , is a weighted function from some Muckenhoupt class, and , are weighted capacity and measure of the corresponding sets.
Cet exposé présente les résultats de l’article [3] au sujet des ondes progressives pour l’équation de Gross-Pitaevskii : la construction d’une branche d’ondes progressives non constantes d’énergie finie en dimensions deux et trois par un argument variationnel de minimisation sous contraintes, ainsi que la non-existence d’ondes progressives non constantes d’énergie petite en dimension trois.
We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space [...] W˙2,1p,φ(Q,ω).
We prove pointwise gradient bounds for entire solutions of pde’s of the form ℒu(x) = ψ(x, u(x), ∇u(x)), where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.
Let be a weak solution of a quasilinear elliptic equation of the growth with a measure right hand term . We estimate at an interior point of the domain , or an irregular boundary point , in terms of a norm of , a nonlinear potential of and the Wiener integral of . This quantifies the result on necessity of the Wiener criterion.