scholarly journals Absolute continuity of parabolic measure and the initial-dirichlet problem

2019 ◽  
Author(s):  
◽  
Alyssa Genschaw
Keyword(s):  
Dirichlet Problem ◽  
Absolute Continuity ◽  
Divergence Form ◽  
Surface Measure ◽  
Scale Invariant ◽  
Elliptic Case ◽  
Open Set ◽  
Caloric Measure ◽  
Reverse Hölder ◽  
Hölder Estimate

This thesis is devoted to the study of parabolic measure corresponding to a divergence form parabolic operator. We first extend to the parabolic setting a number of basic results that are well known in the elliptic case. Then following a result of Bennewitz-Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set [omega] R(n+1), assuming as a background hypothesis only that the essential boundary of [omega] satisfies an appropriate parabolic version of Ahlfors-David regularity (which entails some backwards in time thickness). We then show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with "lateral" data in [Lp], for some p less than [infinity]. Finally, we prove that for the heat equation, BMO-solvability implies scale invariant quantitative absolute continuity of caloric measure with respect to surface measure, in an open set [omega] with time-backwards ADR boundary. Moreover, the same results apply to the parabolic measure associated to a uniformly parabolic divergence form operator (L), with estimates depending only on dimension, the ADR constants, and parabolicity, provided that the continuous Dirichlet problem is solvable for (L) in [omega]. By a result of Fabes, Garofalo and Lanconelli [FGL], this includes the case of [C1]-Dini coefficients.

scholarly journals Quasilinear elliptic systems in divergence form associated to general nonlinearities

2018 ◽  
Vol 7 (4) ◽  
pp. 425-447 ◽  
Author(s):  
Lorenzo D’Ambrosio ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Elliptic Systems ◽  
Divergence Form ◽  
Systems Of Equations ◽  
Quasilinear Elliptic Systems ◽  
Open Set ◽  
Priori Estimates ◽  
Quasilinear Elliptic

AbstractThe paper is concerned with a priori estimates of positive solutions of quasilinear elliptic systems of equations or inequalities in an open set of {\Omega\subset\mathbb{R}^{N}} associated to general continuous nonlinearities satisfying a local assumption near zero. As a consequence, in the case {\Omega=\mathbb{R}^{N}}, we obtain nonexistence theorems of positive solutions. No hypotheses on the solutions at infinity are assumed.

A Quasi-Linear Elliptic Boundary Value Problem

1966 ◽  
Vol 18 ◽  
pp. 1105-1112 ◽  
Author(s):  
R. A. Adams
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Classical Solutions ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Image Position ◽  
Linear Elliptic Boundary ◽  
Typical Point

Let Ω be a bounded open set in Euclidean n-space, En. Let α = (α1, … , an) be an n-tuple of non-negative integers;and denote by Qm the set ﹛α| 0 ⩽ |α| ⩽ m}. Denote by x = (x1, … , xn) a typical point in En and putIn this paper we establish, under certain circumstances, the existence of weak and classical solutions of the quasi-linear Dirichlet problem1

Perturbation and Solvability of Initial Lp Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains

2014 ◽  
Vol 66 (2) ◽  
pp. 429-452 ◽  
Author(s):  
Jorge Rivera-Noriega
Keyword(s):  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Carleson Measure ◽  
Divergence Form ◽  
Second Order ◽  
Linear Operators ◽  
Dirichlet Problems ◽  
Small Perturbations ◽  
Cylindrical Domains

AbstractFor parabolic linear operators L of second order in divergence form, we prove that the solvability of initial Lp Dirichlet problems for the whole range 1 < p < ∞ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of L satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of p > 1, the initial Lp Dirichlet problem associated with Lu = 0 over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

scholarly journals The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Alberto Cialdea ◽  
Vita Leonessa ◽  
Angelica Malaspina
Keyword(s):  
Dirichlet Problem ◽  
Differential Operators ◽  
Differential Forms ◽  
Variable Coefficients ◽  
Divergence Form ◽  
Boundary Integral ◽  
Second Order ◽  
Boundary Integral Method ◽  
Layer Potential ◽  
Partial Differential Operators

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

scholarly journals Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Najoua Gamara ◽  
Abdelhalim Hasnaoui ◽  
Akrem Makni
Keyword(s):  
Riemannian Manifold ◽  
Dirichlet Problem ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Torsional Rigidity ◽  
Hölder Inequality ◽  
Reverse Hölder Inequality ◽  
Curvature Bounds ◽  
Reverse Hölder

AbstractIn this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains

The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group

2019 ◽  
Vol 21 (01) ◽  
pp. 1750069 ◽  
Author(s):  
Hairong Liu ◽  
Tian Long ◽  
Xiaoping Yang
Keyword(s):  
Dirichlet Problem ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Polynomial Growth ◽  
Bounded Solution ◽  
Type Theorem ◽  
Divergence Form ◽  
Liouville Type ◽  
Periodic Coefficients ◽  
Liouville Type Theorem

We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.

scholarly journals On Dirichlet's boundary value problem for some formally hypoelliptic differntial operators

1988 ◽  
Vol 44 (3) ◽  
pp. 324-349 ◽  
Author(s):  
Niels Jacob
Keyword(s):  
Hilbert Space ◽  
Dirichlet Problem ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Differential Operators ◽  
Boundary Value ◽  
Divergence Form ◽  
Alternative Theorem ◽  
Sesquilinear Form ◽  
Homogeneous Dirichlet Problem

AbstractFor a class of formally hypoelliptic differential operators in divergence form we prove a generalized Gårding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.

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