scholarly journals AnLp-Estimate for Weak Solutions of Elliptic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Sara Monsurrò ◽  
Maria Transirico
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
A Priori ◽  
Unbounded Domains ◽  
Discontinuous Coefficients ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
A Priori Bound

We prove anLp-a priori bound,p>2, for solutions of second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

scholarly journals A Priori Bounds in and in for Solutions of Elliptic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Sara Monsurrò ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
A Priori ◽  
Unbounded Domains ◽  
Variational Problems ◽  
Discontinuous Coefficients ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
A Priori Bounds ◽  
A Priori Bound

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an -a priori bound, . Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.

Some uniqueness results for singular elliptic problems

2015 ◽  
Vol 26 (03) ◽  
pp. 1550026 ◽  
Author(s):  
L. Caso ◽  
R. D'Ambrosio
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Weighted Sobolev Spaces ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
Dirichlet Problems ◽  
Uniqueness Results ◽  
Singular Data

We prove some uniqueness results for Dirichlet problems for second-order linear elliptic partial differential equations in non-divergence form with singular data in suitable weighted Sobolev spaces, on an open subset Ω of ℝn, n ≥ 2, not necessarily bounded or regular.

scholarly journals Nonlinear parabolic problems in unbounded domains

1979 ◽  
Vol 82 (3-4) ◽  
pp. 201-209 ◽  
Author(s):  
Guy Mahler
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Unbounded Domains ◽  
Parabolic Problems ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Nonlinear Parabolic Problems ◽  
Existence Of Weak Solutions ◽  
Nonlinear Parabolic

We show the existence of weak solutions of nonlinear parabolic partial differential equations in unbounded domains, provided that a variant of the Leray-Lions conditions is satisfied.

scholarly journals On Maximum Principles for -Metaharmonic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
A. Mareno
Keyword(s):  
Partial Differential Equations ◽  
Maximum Principle ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
A Priori ◽  
Elliptic Partial Differential Equations ◽  
Maximum Principles ◽  
A Priori Bounds ◽  
Even Order ◽  
Homogeneous Linear

We study homogeneous linear elliptic partial differential equations of even order. Several maximum principle results are deduced for such equations as well as a priori bounds for certain boundary value problems.

scholarly journals W2,p-Solvability of the Dirichlet Problem for Elliptic Equations with Singular Data

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Loredana Caso ◽  
Roberta D’Ambrosio ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Unique Solvability ◽  
Elliptic Equations ◽  
Weighted Sobolev Spaces ◽  
Elliptic Partial Differential Equations ◽  
Planar Case ◽  
Singular Data

We give an overview on some results concerning the unique solvability of the Dirichlet problem inW2,p,p>1, for second-order linear elliptic partial differential equations in nondivergence form and with singular data in weighted Sobolev spaces. We also extend such results to the planar case.

Higher-order estimates for fully nonlinear difference equations. I

2000 ◽  
Vol 43 (3) ◽  
pp. 485-510 ◽  
Author(s):  
Derek W. Holtby
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Equations ◽  
A Priori ◽  
Elliptic Partial Differential Equations ◽  
Uniform Mesh ◽  
Partial Differential ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Special Case

AbstractThe purpose of this work is to establish a priori C2, α estimates for mesh function solutions of nonlinear positive difference equations in fully nonlinear form on a uniform mesh, where the fully nonlinear finite-difference operator ℱh is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We deal here with the special case that the operator does not depend explicitly upon the independent variables. We do this by discretizing the approach of Evans for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger. The result in this special case forms the basis for a more general result in part II. We also derive the discrete interpolation inequalities needed to obtain estimates for the interior C2, α semi-norm in terms of the C0 norm.

scholarly journals On the Cauchy problem for semilinear elliptic equations

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Nguyen Huy Tuan ◽  
Tran Thanh Binh ◽  
Tran Quoc Viet ◽  
Daniel Lesnic
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Elliptic Equations ◽  
Elliptic Partial Differential Equations ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
The Cauchy Problem ◽  
Ill Posed

AbstractWe study the Cauchy problem for nonlinear (semilinear) elliptic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak

scholarly journals ON ASYMPTOTIC PROPERTIES OF SOLUTIONS, DEFINED ON THE HALF OF AXIS OF ONE SEMILINEAR ODE

2017 ◽  
Vol 21 (6) ◽  
pp. 130-134
Author(s):  
I.V. Filimonova ◽  
T.S. Khachlaev
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unknown Function ◽  
Linear Equations ◽  
Unbounded Domains ◽  
Elliptic Partial Differential Equations ◽  
Order Derivative ◽  
Partial Differential ◽  
First Order ◽  
Fowler Equation

The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential. The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.

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