On the regularity of optimal potentials in control problems governed by elliptic equations
-
Giuseppe Buttazzo
,
Juan Casado-Díaz
and
Faustino Maestre
Abstract
In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the
Funding statement: The work of Giuseppe Buttazzo is part of the project 2017TEXA3H “Gradient flows, Optimal Transport and Metric Measure Structures” funded by the Italian Ministry of Research and University. Giuseppe Buttazzo is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The work of Juan Casado-Díaz and Faustino Maestre is a part of the FEDER project PID2020-116809GB-I00 of the Ministerio de Ciencia e Innovación of the government of Spain and the project UAL2020-FQM-B2046 of the Consejería de Transformación Económica, Industria, Conocimiento y Universidades of the regional government of Andalusia, Spain.
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- Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity
- On the regularity of optimal potentials in control problems governed by elliptic equations
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Articles in the same Issue
- Frontmatter
- Another proof of the existence of homothetic solitons of the inverse mean curvature flow
- A Weierstrass extremal field theory for the fractional Laplacian
- Minimizing movements for anisotropic and inhomogeneous mean curvature flows
- A singular Yamabe problem on manifolds with solid cones
- Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
- Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity
- Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian
- Hierarchy structures in finite index CMC surfaces
- No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature
- Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data
- Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity
- On the regularity of optimal potentials in control problems governed by elliptic equations
- Sobolev embeddings and distance functions
- Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes
- On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary
- Sobolev contractivity of gradient flow maximal functions
- Discrete approximation of nonlocal-gradient energies
- Symmetry and monotonicity of singular solutions to p-Laplacian systems involving a first order term
- Flat flow solution to the mean curvature flow with volume constraint
