相关论文: Max-Min characterization of the mountain pass ener…

We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain mountain pass solutions of critical and…

We derive a rigorous scaling law for minimizers in a natural version of the regularized Cross-Newell model for pattern formation far from threshold. These energy-minimizing solutions support defects having the same character as what is seen…

We prove that a certain class of elliptic free boundary problems, which includes the Prandtl-Batchelor problem from fluid dynamics as a special case, has two distinct nontrivial solutions for large values of a parameter. The first solution…

The Cahn-Hilliard energy landscape on the torus is explored in the critical regime of large system size and mean value close to $-1$. Existence and properties of a "droplet-shaped" local energy minimizer are established. A standard mountain…

We study the existence of mountain-pass solutions to a potential-free mean-field game system in the whole space $\mathbb R^n$ under the mass-supercritical regime, assuming an aggregating local coupling and a $C^2$ Hamiltonian that is…

We present a variational framework for studying the existence of solutions of a class of elliptic free boundary problems on stratified Lie groups. Using the important monotonicity result in a Non-Euclidean setup, we prove that our solution…

It is established the existence and multiplicity of weak solutions for a class of nonlocal equations involving the fractional laplacian, nonlinearities with critical exponential growth and potentials this is which may change sign. The…

In the present paper, we study a double-phase variable exponent problem which is set up within a variational framework including a singular potential of fractional-Hardy-type. We employ the Mountain-Pass theorem and the strong minimum…

Energy minimality selects among possible configurations of a continuous body with and without cracks those compatible with assigned boundary conditions of Dirichlet-type. Crack paths are described in terms of curvature varifolds so that we…

For a functional $\E$ and a peak selection that picks up a global maximum of $\E$ on varying cones, we study the convergence up to a subsequence to a critical point of the sequence generated by a mountain pass type algorithm. Moreover, by…

Feature selection is one of the most fundamental problems in machine learning. An extensive body of work on information-theoretic feature selection exists which is based on maximizing mutual information between subsets of features and class…

We present a bounded-error quantum algorithm for evaluating Min-Max trees. For a tree of size N our algorithm makes N^{1/2+o(1)} comparison queries, which is close to the optimal complexity for this problem.

The minimum energy path (MEP) is the most probable transition path that connects two equilibrium states of a potential energy landscape. It has been widely used to study transition mechanisms as well as transition rates in the fields of…

We introduce variational problems on Riemannian manifolds with constrained acceleration and derive necessary conditions for normal extremals in the constrained variational problem. The problem consists on minimizing a higher-order energy…

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. In this paper we introduce a discretized version of this theory associated to a discrete Morse function on a…

We consider extensions of differential fields of mappings and obtain a lower energy bound for quasiconformal extension fields in terms of the topological degree. We also consider the related minimization problem for the $q$-harmonic energy,…

We develop a variational technique for some wide classes of nonlinear evolutions. The novelty here is that we derive the main information directly from the corresponding Euler-Lagrange equations. In particular, we prove that not only the…

For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations.…

In this paper we give a positive answer to the conjecture raised by Hajaiej et al. (J. Geom. Anal., 2024, 34(6): No. 182, 44 pp) on the existence of a mountain pass solution at positive energy level to the Br\'{e}zis-Nirenberg problem with…

We define the minimum energy state while the expectation value of the field, evolves in time. We obtain the relation between the n-point functions in such a state, and the external field for all the moments. We obtain an equation of motion…