相关论文: Minimizers of $L^{2}$-critical inhomogeneous varia…
In this paper, we study the existence and the concentration behavior of minimizers for $i_V(c)=\inf\limits_{u\in S_c}I_V(u)$, here $S_c=\{u\in H^1(\R^N)|~\int_{\R^N}V(x)|u|^2<+\infty,~|u|_2=c>0\}$ and…
We investigate the existence of local minimizers with prescribed $L^2$-norm for the energy functional associated to the mass-supercritical nonlinear Schr\"{o}dinger equation on the product space $\mathbb{R}^N \times M^k$, where $(M^k,g)$ is…
We study a class of semi-linear problems involving the fractional Laplacian under subcritical or critical growth assumptions. We prove that, for the corresponding functional, local minimizers with respect to a C^0-topology weighted with a…
In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with $A_2$ weights. We show existence and boundedness of minimizers. The key novelty is a sharp $C^{1+\gamma}$ regularity…
We study existence of minimizers of the least gradient problem \[\inf_{v \in BV_g} \int_{\Omega}\varphi(x, Dv),\] where $BV_g=\{v \in BV(\Omega): \int_{\partial \Omega}gv=1\}$, $\varphi(x,p): \Omega\times \R^n \rightarrow \R$ is a convex,…
We consider a nonlocal isoperimetric problem defined in the whole space $\R^N$, whose nonlocal part is given by a Riesz potential with exponent $\alpha\in(0, N-1)$. We show that critical configurations with positive second variation are…
We formulate a variational principle for a collection of projectors in an indefinite inner product space. The existence of minimizers is proved in various situations.
We will study an open problem pertaining to the uniqueness of minimizers for a class of variational problems emanating from Meyer's model for the decomposition of an image into a geometric part and a texture part. Mainly, we are interested…
We study a variational problem motivated by models of species population density in a nonhomogeneous environment. We first analyze local minimizers and the structure of the saturated region (where the population attains its maximal density)…
This paper considers ground states of mass subcritical rotational nonlinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u+i\Omega(x^\perp\cdot\nabla u)=\mu u+\rho^{p-1}|u|^{p-1}u \,\ \text{in} \,\ \mathbb{R}^2, \end{equation*}…
We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional nonlinear Schr\"odinger equation with periodic potentials. Using the concentration-compactness principle, we show a…
We study variational problems for second order supremal functionals $\mathrm F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm L^{\infty}(\Omega)}$, where $F$ satisfies certain natural assumptions, $\mathrm A$ is…
In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla…
We study existence of minimisers to the least gradient problem on a strictly convex domain in two settings. On a bounded domain, we allow the boundary data to be discontinuous and prove existence of minimisers in terms of the Hausdorff…
We study the planar least gradient problem with respect to an anisotropic norm $\phi$ for continuous boundary data. We prove existence of minimizers for strictly convex domains $\Omega$. Furthermore, we inspect the issue of uniqueness and…
We consider the problem: $$\inf_{{u}\in {H}^{1}_{g}(\Omega),\|u\|_{q}=1} \int_{\Omega}{p(x)}|\nabla{u(x)}|^{2}dx-\lambda\int_{\Omega}| u(x)|^{2}dx$$ where $\Omega$ is a bounded domain in $\R^{n}$, ${n}\geq{4}$, $ p :…
In this paper, we investigate the constrained minimization problem \begin{equation}\label{eq:0.1} e(a):=\inf_{\{u\in \mathcal{H},\|u\|_2^2=1\}}E_a(u), \end{equation} where the energy functional \begin{equation} \label{eq:0.2}…
In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincar\'{e}-Sobolev inequalities. We show that minimizers do exist for smooth domains in $\mathbb{R}^d$, an also for some polyhedral domains. On…
In this paper, we consider complex-valued solutions of the planar Schr\"odinger-Newton system, which can be described by minimizers of the constraint minimization problem. It is shown that there exists a critical rotational velocity…
We consider the question of quantitative stability of minimisers for a well-known variational problem for which the infimum of the energy is not achieved in the classical sense, namely for the Dirichlet energy of degree $1$ maps from closed…