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Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces

Vagif S. Guliyev, Mehriban N. Omarova (2016)

Open Mathematics

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) W ˙ 2 , 1 p , ϕ Q , ω .

Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s

Alberto Farina, Enrico Valdinoci (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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.

Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points

Jan Malý (1996)

Commentationes Mathematicae Universitatis Carolinae

Let u be a weak solution of a quasilinear elliptic equation of the growth p with a measure right hand term μ . We estimate u ( z ) at an interior point z of the domain Ω , or an irregular boundary point z Ω , in terms of a norm of u , a nonlinear potential of μ and the Wiener integral of 𝐑 n Ω . This quantifies the result on necessity of the Wiener criterion.

Positive solutions of nonlinear elliptic systems

Robert Dalmasso (1993)

Annales Polonici Mathematici

We study the existence and nonexistence of positive solutions of nonlinear elliptic systems in an annulus with Dirichlet boundary conditions. In particular, L a priori bounds are obtained. We also study a general multiple linear eigenvalue problem on a bounded domain.

Prescribing a fourth order conformal invariant on the standard sphere, part II : blow up analysis and applications

Zindine Djadli, Andrea Malchiodi, Mohameden Ould Ahmedou (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere ( 𝕊 n , h ) . We derive from this analysis some a priori estimates in dimension 5 and 6 . On 𝕊 5 these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On 𝕊 6 we prove the existence of at least one...

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