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$H^{2}$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $π$

Abdelkader Tami, Mounir Tlemcani (2021)

Applications of Mathematics

We consider a biharmonic problem $Δ^{2} u_{ω} = f_{ω}$ with Navier type boundary conditions $u_{ω} = Δ u_{ω} = 0$ , on a family of truncated sectors $Ω_{ω}$ in $ℝ^{2}$ of radius $r$ , $0 < r < 1$ and opening angle $ω$ , $ω \in (2 π / 3, π]$ when $ω$ is close to $π$ . The family of right-hand sides ${(f_{ω})}_{ω \in (2 π / 3, π]}$ is assumed to depend smoothly on $ω$ in $L^{2} (Ω_{ω})$ . The main result is that $u_{ω}$ converges to $u_{π}$ when $ω \to π$ with respect to the $H^{2}$ -norm. We can also show that the $H^{2}$ -topology is optimal for such a convergence result.

Harmonic Mappings and Minimal Submanifolds.

J. Jost, S. Hildebrandt, K.-O. Widman (1980/1981)

Inventiones mathematicae

Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations

Tuomo Kuusi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.

Harnack inequalities and ABP estimates for nonlinear second-order elliptic equations in unbounded domains.

Amendola, M.E., Rossi, L., Vitolo, A. (2008)

Abstract and Applied Analysis

Harnack Inequalities for Nonuniformly Elliptic Divergence Structure Equations.

Neil S. Trudinger (1981)

Inventiones mathematicae

Harnack inequalities for quasi-minima of variational integrals

E. Di Benedetto, Neil S. Trudinger (1984)

Annales de l'I.H.P. Analyse non linéaire

Harnack inequalities for Schrödinger operators

Wolfhard Hansen (1999)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Harnack inequalities on graphs

Thierry Delmotte (1997/1998)

Séminaire de théorie spectrale et géométrie

Harnack inequality and Green function for a certain class of degenerate elliptic differential operators.

Oscar Salinas (1991)

Revista Matemática Iberoamericana

The main purpose of this work is to obtain a Harnack inequality and estimates for the Green function for the general class of degenerate elliptic operators described below.

Harnack inequality for the Schrödinger problem relative to strongly local Riemannian $p$ -homogeneous forms with a potential in the Kato class.

Biroli, Marco, Marchi, Silvana (2007)

Boundary Value Problems [electronic only]

Harnack type estimates for nonlinear elliptic systems and applications

Jérôme Busca, Boyan Sirakov (2004)

Annales de l'I.H.P. Analyse non linéaire

Harnack's estimates: positivity and local behavior of degenerate and singular parabolic equations.

DiBenedetto, Emmanuele, Gianazza, Ugo, Safonov, Mikhail, Urbano, José Miguel, Vespri, Vincenzo (2007)

Boundary Value Problems [electronic only]

Harnack’s inequalities for solutions to the mean curvature equation and to the capillarity problem

Fei-Tsen Liang (2003)

Rendiconti del Seminario Matematico della Università di Padova

Harnack's inequality for parabolic operators with singular low order terms.

Karl-Theodor Sturm (1994)

Mathematische Zeitschrift

Harnack's inequality for quasilinear elliptic equations with coefficients in Morrey spaces

Pietro Zamboni (1993)

Rendiconti del Seminario Matematico della Università di Padova

Harnack's inequality for solutions of some degenerate elliptic equations.

Ahmed Mohammed (2002)

Revista Matemática Iberoamericana

We prove Harnack's inequality for non-negative solutions of some degenerate elliptic operators in divergence form with the lower order term coefficients satisfying a Kato type contition.

High frequency asymptotic solutions of the reduced wave equation on infinite regions with nonconvex boundaries.

Bloom, Clifford O. (1996)

Mathematical Problems in Engineering

High-electric-field limit for the Vlasov–Maxwell–Fokker–Planck system

Mihai Bostan, Thierry Goudon (2008)

Annales de l'I.H.P. Analyse non linéaire

Higher regularity for nonlinear oblique derivative problems in Lipschitz domains

Gary M. Lieberman (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

There is a long history of studying nonlinear boundary value problems for elliptic differential equations in a domain with sufficiently smooth boundary. In this paper, we show that the gradient of the solution of such a problem is continuous when a directional derivative is prescribed on the boundary of a Lipschitz domain for a large class of nonlinear equations under weak conditions on the data of the problem. The class of equations includes linear equations with fairly rough coefficients as well...

Higher regularity of weak solutions of strongly nonlinear elliptic equations

Simader, Christian G. (1986)

Equadiff 6

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