Related papers: Degenerate elliptic equations with $\Phi$-admissib…
We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\Omega\subset\mathbb{R}^n$, let $\rho$ be a quasi-metric on $\Omega$,…
Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius.…
We consider an elliptic problem with unknowns on the boundary of the domain of the elliptic equation and suppose that the right-hand side of this equation is square integrable and that the boundary data are arbitrary (specifically,…
The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying…
We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty, T) \times \mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb R^d_+$, where $\mathbb…
We prove the Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate local behaviors of solutions of such equations by developing local boundedness and weak Harnack…
The main result of the paper is on the continuity of weak solutions of infinitely degenerate quasilinear second order equations. Namely, we show that every weak solution to a certain class of degenerate quasilinear equations is continuous.…
As shown in [15], under some structural assumptions, working on congested traffic problems in general and increasingly dense networks leads, at the limit by {\Gamma}-convergence, to continuous minimization problems posed on measures on…
We establish Holder continuity of weak solutions to degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type.
The main goal of this paper is twofold. First, it characterizes the existence of positive periodic solutions for a general class of weighted periodic-parabolic logistic problems of degenerate type (see (1.1)). This result provides us with…
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to H\"{o}rmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We…
By borrowing ideas from the parabolic theory, we use a combination of De Giorgi's and Moser's methods to give some remarks on the proof of H\"older continuity of weak solutions of elliptic equations.
Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE $-\mathrm{div}\left((\vert\nabla…
We use De Giorgi-Nash-Moser iteration scheme to establish that weak solutions to a coupled system of elliptic equations with critical growth on the boundary are in $L^\infty(\Omega)$. Moreover, we provide an explicit $L^\infty(\Omega)$-…
As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in…
We establish the existence and sharp global regularity results ($C^{0, \gamma}$, $C^{0, 1}$ and $C^{1, \alpha}$ estimates) for a class of fully nonlinear elliptic PDEs with unbalanced variable degeneracy. In a precise way, the degeneracy…
We prove a Poincar\'e-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical…
We study the boundary regularity properties and derive a priori pointwise supremum estimates of weak solutions and their derivatives in terms of suitable weighted $L^2$-norms for a class of degenerate parabolic equations that satisfy…
We investigate the regularity of local weak solutions to evolution equations of the form \[…
The regularity theory of the degenerate complex Monge-Amp\`{e}re equation is studied. The equation is considered on a closed compact K\"{a}hler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a…