Existence results for dirichlet problems in L1via minty's lemma

2000 ◽  
Vol 76 (3-4) ◽  
pp. 309-317 ◽  
Author(s):  
Lucio Boccardo ◽  
Luigi Orsina
Keyword(s):  
Existence Results ◽  
Dirichlet Problems ◽  
Minty’S Lemma

Symmetry of solutions to optimization problems related to partial differential equations

2006 ◽  
Vol 136 (5) ◽  
pp. 921-934 ◽  
Author(s):  
Fabrizio Cuccu ◽  
Giovanni Porru
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Optimization Problems ◽  
Existence Results ◽  
Dirichlet Problems ◽  
Partial Differential ◽  
Moving Plane Method ◽  
Plane Method ◽  
Moving Plane

We investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving-plane method to find symmetry results for solutions of a system. We apply these results in our discussion of symmetry for the maximal configurations of the previous problem.

scholarly journals Semilinear systems with a multi-valued nonlinear term

2017 ◽  
Vol 15 (1) ◽  
pp. 628-644
Author(s):  
In-Sook Kim ◽  
Suk-Joon Hong
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Nonlinear Term ◽  
Degree Theory ◽  
Existence Results ◽  
Topological Degree Theory ◽  
Dirichlet Problems ◽  
Semilinear Systems ◽  
Densely Defined ◽  
Monotone Type

Abstract Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = (N1, N2), provided that L = (L1, L2) satisfies dim Ker L1 = ∞ and dim Ker L2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed.

On the Dirichlet problem with nonconvex nonlinearity

2010 ◽  
Vol 47 (2) ◽  
pp. 190-199
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Growth Conditions ◽  
Existence Results ◽  
Dirichlet Problems ◽  
Local Growth ◽  
Dual Variational Method ◽  
Nonlinear Eigenvalue

We provide existence results for 2 m order Dirichlet problems with nonconvex nonlinearity which satisfies general local growth conditions. In doing so we construct a dual variational method. Problem considered relates to the problem of nonlinear eigenvalue.

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