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Penalization of Dirichlet optimal control problems

Eduardo Casas, Mariano Mateos, Jean-Pierre Raymond (2009)

ESAIM: Control, Optimisation and Calculus of Variations

We apply Robin penalization to Dirichlet optimal control problems governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.

Penalization of Dirichlet optimal control problems

Eduardo Casas, Mariano Mateos, Jean-Pierre Raymond (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We apply Robin penalization to Dirichlet optimal control problems governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.

Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

Tomasz Klimsiak, Andrzej Rozkosz (2016)

Colloquium Mathematicae

We are mainly concerned with equations of the form -Lu = f(x,u) + μ, where L is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, f satisfies the monotonicity condition and mild integrability conditions, and μ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric...

Semilinear elliptic problems in unbounded domains

Aleksandra Orpel (2006)

Applicationes Mathematicae

We investigate the existence of positive solutions and their continuous dependence on functional parameters for a semilinear Dirichlet problem. We discuss the case when the domain is unbounded and the nonlinearity is smooth and convex on a certain interval only.

Singularly perturbed elliptic equations with solutions concentrating on a 1-dimensional orbit

Bernhard Ruf, P.N. Srikanth (2010)

Journal of the European Mathematical Society

We consider a singularly perturbed elliptic equation with superlinear nonlinearity on an annulus in 4 , and look for solutions which are invariant under a fixed point free 1-parameter group action. We show that this problem can be reduced to a nonhomogeneous equation on a related annulus in dimension 3. The ground state solutions of this equation are single peak solutions which concentrate near the inner boundary. Transforming back, these solutions produce a family of solutions which concentrate...

Stability of the Pohožaev obstrucion in dimension 3

Olivier Druet, Paul Laurain (2010)

Journal of the European Mathematical Society

We investigate problems connected to the stability of the well-known Pohoˇzaev obstruction. We generalize results which were obtained in the minimizing setting by Brezis and Nirenberg [2] and more recently in the radial situation by Brezis and Willem [3].

Stable solutions of Δ u = f ( u ) in N

Louis Dupaigne, Alberto Farina (2010)

Journal of the European Mathematical Society

Several Liouville-type theorems are presented for stable solutions of the equation - Δ u = f ( u ) in N , where f > 0 is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

Symmetry of local minimizers for the three-dimensional Ginzburg–Landau functional

Vincent Millot, Adriano Pisante (2010)

Journal of the European Mathematical Society

We classify nonconstant entire local minimizers of the standard Ginzburg–Landau functional for maps in H loc 1 ( 3 ; 3 ) satisfying a natural energy bound. Up to translations and rotations,such solutions of the Ginzburg–Landau system are given by an explicit solution equivariant under the action of the orthogonal group.

Currently displaying 61 – 80 of 83