scholarly journals Local regularity for concave homogeneous complex degenerate elliptic equations dominating the Monge–Ampère equation

2021 ◽  
Author(s):  
Soufian Abja ◽  
Guillaume Olive
Keyword(s):  
Elliptic Equations ◽  
Regularity Result ◽  
Nonlinear Elliptic Equations ◽  
Local Regularity ◽  
Degenerate Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Homogeneous Complex ◽  
Degenerate Elliptic ◽  
Monge Ampere

AbstractIn this paper, we establish a local regularity result for $$W^{2,p}_{{\mathrm {loc}}}$$ W loc 2 , p solutions to complex degenerate nonlinear elliptic equations $$F(D^2_{\mathbb {C}}u)=f$$ F ( D C 2 u ) = f when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called k-Monge–Ampère equation.

scholarly journals Existence Results For A Class Of Nonlinear Degenerate Elliptic Equations

2019 ◽  
Vol 5 (2) ◽  
pp. 164-178
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Nonlinear Elliptic Equations ◽  
Existence Results ◽  
Degenerate Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
Nonlinear Degenerate

AbstractIn this paper we are interested in the existence of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations\left\{ {\matrix{ { - {\rm{div}}\left[ {\mathcal{A}\left( {x,\nabla u} \right){\omega _1} + \mathcal{B}\left( {x,u,\nabla u} \right){\omega _2}} \right] = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)\,\,{\rm{in}}} \,\,\,\,\,\Omega ,} \hfill \cr {u\left( x \right) = 0\,\,\,\,{\rm{on}}\,\,\,\,\partial \Omega {\rm{,}}} \hfill \cr } } \right.in the setting of the weighted Sobolev spaces.

scholarly journals ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains

2016 ◽  
Vol 18 (04) ◽  
pp. 1550075 ◽  
Author(s):  
I. Birindelli ◽  
I. Capuzzo Dolcetta ◽  
A. Vitolo
Keyword(s):  
Elliptic Equations ◽  
Unbounded Domains ◽  
Nonlinear Elliptic Equations ◽  
Degenerate Elliptic Equations ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Geometric Conditions ◽  
Degenerate Elliptic ◽  
Hölder Estimates

We prove global Hölder estimates for solutions of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions à la Cabré.

scholarly journals Existence of a weak bounded solution for nonlinear degenerate elliptic equations in Musielak-Orlicz spaces

2020 ◽  
Vol 6 (1) ◽  
pp. 16-33 ◽  
Author(s):  
M. Bourahma ◽  
J. Bennouna ◽  
M. El Moumni
Keyword(s):  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Bounded Solution ◽  
Orlicz Spaces ◽  
Nonlinear Elliptic Equations ◽  
Degenerate Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
Nonlinear Degenerate

AbstractIn this paper, we show the existence of solutions for the nonlinear elliptic equations of the form\left\{ {\matrix{ { - {\rm{div}}\,a\left( {x,u,\nabla u} \right) = f,} \hfill \cr {u \in W_0^1L\varphi \left( \Omega \right) \cap {L^\infty }\left( \Omega \right),} \hfill \cr } } \right.where a\left( {x,s,\xi } \right) \cdot \xi \ge \bar \varphi _x^{ - 1}\left( {\varphi \left( {x,h\left( {\left| s \right|} \right)} \right)} \right)\varphi \left( {x,\left| \xi \right|} \right) and h : ℝ+→]0, 1] is a continuous decreasing function with unbounded primitive. The second term f belongs to LN(Ω) or to Lm(Ω), with m = {{rN} \over {r + 1}} for some r > 0 and φ is a Musielak function satisfying the Δ2-condition.

scholarly journals On degenerate fully nonlinear elliptic equations in balls

1987 ◽  
Vol 35 (2) ◽  
pp. 299-307 ◽  
Author(s):  
Neil S. Trudinger
Keyword(s):  
Elliptic Equations ◽  
Existence Theorems ◽  
Nonlinear Elliptic Equations ◽  
Fully Nonlinear Elliptic Equations ◽  
Neumann Problems ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
Monge Ampere ◽  
Special Case ◽  
Nonlinear Degenerate

We establish derivative estimates and existence theorems for the Dirichlet and Neumann problems for nonlinear, degenerate elliptic equations of the form F (D2u) = g in balls. The degeneracy arises through the possible vanishing of the function g and the degenerate Monge-Ampère equation is covered as a special case.

On the viscosity solutions of eigenvalue problems for a class of nonlinear elliptic equations

2019 ◽  
Vol 12 (4) ◽  
pp. 393-421
Author(s):  
Tilak Bhattacharya ◽  
Leonardo Marazzi
Keyword(s):  
Viscosity Solutions ◽  
Elliptic Equations ◽  
Eigenvalue Problems ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
First Eigenvalue ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
First Eigenfunction ◽  
Nonlinear Degenerate

AbstractWe consider viscosity solutions of a class of nonlinear degenerate elliptic equations, involving a parameter, on bounded domains. These arise in the study of eigenvalue problems. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of the first eigenvalue and an associated positive first eigenfunction.

scholarly journals An oscillation theorem for the nonlinear degenerate elliptic equation in the Heisenberg group

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Duan Wu ◽  
Pengcheng Niu
Keyword(s):  
Elliptic Equation ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Degenerate Elliptic Equation ◽  
Oscillation Theorem ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
Oscillation Of Solutions ◽  
Nonlinear Degenerate

AbstractThe aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group $H^{n}$ H n . We first derive a critical inequality in $H^{n}$ H n . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in $R^{n}$ R n .

Sign in / Sign up

Export Citation Format

Share Document