Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Guy Barles; Emmanuel Chasseigne; Cyril Imbert

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Displaying similar documents to “Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations”

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Dario Daniele Monticelli (2010)

Journal of the European Mathematical Society

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We deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for...

$L^{p, Φ}$ spaces and their application to elliptic partial differential operators

Magdalena Jaroszewska (1983)

Annales Polonici Mathematici

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Partially elliptic pseudodifferential operators and the $W F_{x}$ of distributions

Annalisa Menegus, Giuseppe Olivi (1976)

Rendiconti del Seminario Matematico della Università di Padova

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The Calderón-Zygmund theory for elliptic problems with measure data

Giuseppe Mingione (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider non-linear elliptic equations having a measure in the right-hand side, of the type $div a (x, D u) = μ,$ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori...

Estimates on elliptic equations that hold only where the gradient is large

Cyril Imbert, Luis Silvestre (2016)

Journal of the European Mathematical Society

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We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the Hölder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.

Wiener criterion for degenerate elliptic obstacle problem

Marco Biroli, Umberto Mosco (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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We give a Wiener criterion for the continuity of an obstacle problem relative to an elliptic degenerate problem with a weight in the $A_{2}$ class.

On a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

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We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u |}_{\partial Ω} = 0$ , where f(x) and h(u) satisfy minimal regularity assumptions.

Levi's parametrix for some sub-elliptic non-divergence form operators.

Bonfiglioli, Andrea, Lanconelli, Ermanno, Uguzzoni, Francesco (2003)

Electronic Research Announcements of the American Mathematical Society [electronic only]

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Wiener criterion for degenerate elliptic obstacle problem

Marco Biroli, Umberto Mosco (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We give a Wiener criterion for the continuity of an obstacle problem relative to an elliptic degenerate problem with a weight in the $A_{2}$ class.

On the principal eigenvalue of elliptic operators in $ℝ^{N}$ and applications

Henry Berestycki, Luca Rossi (2006)

Journal of the European Mathematical Society

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Two generalizations of the notion of principal eigenvalue for elliptic operators in $ℝ^{N}$ are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and “limit periodic” operators. These results apply to questions of existence and uniqueness for some semilinear problems in the whole space. We also indicate several outstanding open problems and formulate some conjectures.

A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

Banach Center Publications

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The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $- \sum_{i, j = 1}^{n} D_{i} (a_{i j} D_{j} u) = f (u, \int_{Ω} g (u^{p}))$ , in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

Generating singularities of solutions of quasilinear elliptic equations using Wolff’s potential

Darko Žubrinić (2003)

Czechoslovak Mathematical Journal

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We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of $p$ -Laplacian type. If $p < γ < N$ and the right-hand side is a Radon measure with singularity of order $γ$ at $x_{0} \in Ω$ , then any supersolution in $W_{l o c}^{1, p} (Ω)$ has singularity of order at least $\frac{(γ - p)}{(p - 1)}$ at $x_{0}$ . In the proof we exploit a pointwise estimate of $𝒜$ -superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff’s potential of Radon’s measure.

A characterization of elliptic operators

Grzegorz Łysik, Paweł M. Wójcicki (2014)

Annales Polonici Mathematici

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We give a characterization of constant coefficients elliptic operators in terms of estimates of their iterations on smooth functions.

Optimal $L^{p}$ -properties of Green’s functions for non-divergence elliptic equations in two dimensions

Gioconda Moscariello, Carlo Sbordone (2005)

Studia Mathematica

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A sharp integrability result for non-negative adjoint solutions to planar non-divergence elliptic equations is proved. A uniform estimate is also given for the Green's function.

A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Guillermo Reyes, Juan Luis Vázquez (2006)

Journal of the European Mathematical Society

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In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{*}$ , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the...

Uniqueness of solutions for some degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2014)

Applicationes Mathematicae

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We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $- \sum_{i, j = 1}^{n} D_{j} (a_{i j} (x) D_{i} u (x)) + b (x) u (x) + d i v (Φ (u (x))) = g (x) - \sum_{j = 1}^{n} f_{j} (x)$ on Ω in the setting of the space H₀(Ω).

Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity $q > 1$ and with natural growth

Sofia Giuffrè, Giovanna Idone (2005)

Bollettino dell'Unione Matematica Italiana

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In this paper we deal with the Hölder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second order discontinuous elliptic systems with nonlinearity $q > 1$ and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global Hölder continuous in the case of the dimension $n = q$ without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets are always...