Article
Mathematics, Applied
Harald Garcke, Patrik Knopf, Sourav Mitra, Anja Schlomerkemper
Summary: In this article, the strong well-posedness, stability and optimal control of a two-dimensional incompressible magneto-viscoelastic fluid model are studied. Global strong solution in a suitable functional framework is proved to exist for the model. Stability estimates with respect to an external magnetic field are derived, and the external magnetic field is used as the control to minimize a tracking-type cost functional. Existence of an optimal control and first-order necessary optimality conditions are proven. A second optimal control problem is also considered, where the external magnetic field, representing the control, is generated by a finite number of fixed magnetic field coils.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Phan Thanh Nam, Marcin Napiorkowski
Summary: The study focuses on the homogeneous Bose gas on a unit torus in the mean-field regime, deriving a two-term expansion of the one-body density matrix of the ground state as the number of particles becomes large. The proof is based on a cubic correction to Bogoliubov's approximation of the ground state energy and the ground state.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Jose A. Carrillo, Ruiwen Shu
Summary: We establish the radial symmetry of local minimizers in the infinite Wasserstein distance for the interaction energy with repulsive-attractive potentials, under generic conditions. Consequently, we prove the uniqueness of local minimizers in this topology for a certain class of interaction potentials. Introducing a novel notion of concavity of the interaction potential enables us to reveal fractal-like behavior of the local minimizers. We provide a family of interaction potentials that result in local minimizers with no isolated points in their support and no interior points in any superlevel set.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Stefano Biagi, Fabio Punzo
Summary: This paper investigates Liouville-type theorems for elliptic equations with a drift and a potential in bounded domains. It provides sufficient conditions for the equation to not have nontrivial bounded solutions, and shows the optimality of these conditions. In fact, when the conditions fail, the elliptic equation possesses infinitely many bounded solutions.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Weiwei Ao, Aleks Jevnikar, Wen Yang
Summary: We investigate the Sinh-Gordon equation in bounded domains and construct blow up solutions with residual mass exhibiting partial or asymmetric blow up. This is the first result concerning residual mass for this equation, indicating that the concentration-compactness theory of Brezis-Merle with vanishing residuals cannot be extended to this class of problems.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
Guido De Philippis, Luca Spolaor, Bozhidar Velichkov
Summary: This paper proves a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. Additionally, it demonstrates the regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.
INVENTIONES MATHEMATICAE
(2021)
Article
Mathematics, Applied
Gilbert Peralta, Karl Kunisch
Summary: This paper considers a mixed finite element discretization for an optimal control problem of the linear wave equation with homogeneous Dirichlet boundary condition. A Petrov-Galerkin scheme is used for temporal discretization, and Raviart-Thomas finite elements are used for spatial discretization. A priori error analysis is conducted for this numerical scheme. A hybridized formulation is proposed, and it is observed that applying the Arnold-Brezzi post-processing method leads to better convergence rates with respect to space. The interchangeability of discretization and optimization holds for both mixed and hybrid formulations. Numerical experiments are presented to illustrate the theoretical results using the lowest-order Raviart-Thomas elements.
NUMERISCHE MATHEMATIK
(2022)
Article
Mathematics, Applied
Rirong Yuan
Summary: This paper examines the solvability of Dirichlet problems for elliptic equations with different characteristics and discusses the relationship between them and the C-subsolution introduced by Szekelyhidi, as well as the connection to previous studies on gradient estimates. The existence and regularity of admissible solutions to Dirichlet problems for fully nonlinear elliptic equations on compact Kahler manifolds of nonnegative orthogonal bisectional curvature are also established.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
F. S. Albuquerque, J. L. Carvalho, G. M. Figueiredo, E. Medeiros
Summary: This paper investigates the existence of solutions to the planar non-autonomous Schrödinger-Poisson system under the assumption of exponential critical growth of the nonlinearity. A new weighted Trudinger-Moser type inequality is proved, leading to the derivation of a ground state solution to the system.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
J. A. Carrillo, J. Mateu, M. G. Mora, L. Rondi, L. Scardia, J. Verdera
Summary: This paper characterises the minimisers of a one-parameter family of nonlocal and anisotropic energies in probability measures in Rn. It proves the uniqueness of minimisers within a certain range and demonstrates the impact of anisotropy of the interaction kernel on the shape of minimisers, highlighting a paradigmatic example in higher dimensions.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Craig Cowan, Abbas Moameni
Summary: The main objective of this article is to exploit the specific characteristics of a given problem in order to improve compactness for supercritical problems and prove the existence of new types of solutions. The authors introduce an efficient tool in variational methods to construct a new type of classical solutions for a large class of supercritical elliptic PDEs. The issue of symmetry breaking is fundamental in mathematics and physics and the authors establish several symmetry breaking solutions using a variational approach. They also establish new Sobolev embeddings for functions with mild monotonicity on symmetric monotonic domains.
MATHEMATISCHE ANNALEN
(2023)
Article
Operations Research & Management Science
Alberto Dominguez Corella, Nicolai Jork, Vladimir M. Veliov
Summary: The paper investigates the stability properties of solutions of optimal control problems constrained by semilinear parabolic partial differential equations. It obtains Holder or Lipschitz dependence of the optimal solution on perturbations for problems in which the equation and objective functional are affine with respect to the control. The main results are based on an extension of recently introduced assumptions on the growth of the first and second variation of the objective functional and provide stability and error estimates.
COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Birger Brietzke, Hans Knuepfer
Summary: This article investigates the onset of pattern formation in an ultrathin ferromagnetic film of the form Omega(t) := Omega x [0, t]. The relative micromagnetic energy is described by e[M], which represents the energy difference for a given magnetization M with a preferred perpendicular magnetization direction. The scaling of the minimal energy and a BV-bound in the critical regime are derived for the film's base area size dependent pattern formation.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Rustum Choksi, Irene Fonseca, Jessica Lin, Raghavendra Venkatraman
Summary: This paper establishes bounds on the homogenized surface tension for a heterogeneous Allen-Cahn energy functional in a periodic medium. The approach is based on relating the homogenized energy to a purely geometric variational problem involving the large scale behaviour of the signed distance function to a hyperplane in periodic media. Motivated by this, a homogenization result for the signed distance function to a hyperplane in both periodic and almost periodic media is proven.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Luigi De Rosa, Mickael Latocca, Giorgio Stefani
Summary: We prove that the hydrodynamic pressure p associated to the velocity u is an element of C-theta(omega), theta is an element of (0, 1), of an inviscid incompressible fluid in a bounded and simply connected domain omega subset of R-d with C2+ boundary. The pressure p is an element of C-theta (omega) for theta <= 1/2 and p is an element of C-1,C-2 theta-1(omega) for theta > 1/2. We extend and improve the recent result of Bardos and Titi (Philos Trans R Soc A, 2022) obtained in the planar case to every dimension d >= 2 and it also doubles the pressure regularity. When & part;omega is an element of C3+, we prove that an almost double Holder regularity p is an element of C2 theta-(omega) holds even for theta < 1/2. Differently from Bardos and Titi (2022), we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary free case of the d-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
