Volume 11, Issue 1

In this paper we approximate a Kantorovich potential and a transport density for the mass transport problem of two measures (with the transport cost given by a Finsler distance), by taking limits, as p goes to infinity, to a family of variational problems of p -Laplacian type. We characterize the Euler–Lagrange equation associated to the variational Kantorovich problem. We also obtain different characterizations of the Kantorovich potentials and a Benamou–Brenier formula for the transport problem.

In [15], Jean Taylor proved a regularity theorem away from the boundary for Almgren almost minimal sets of dimension 2 in ℝ3{\mathbb{R}^{3}}. It is quite important for understanding the soap films and the solutions of Plateau’s problem away from boundary. In this paper, we will give a regularity result on the boundary for two-dimensional sliding almost minimal sets in ℝ3{\mathbb{R}^{3}}.

Let Ω be a domain in ℝn{\mathbb{R}^{n}}, where n=2,3{n=2,3}. Suppose that a sequence of Sobolev homeomorphisms fk:Ω→ℝn{f_{k}\colon\Omega\to\mathbb{R}^{n}} with positive Jacobian determinants, J⁢(x,fk)>0{J(x,f_{k})>0}, converges weakly in W1,p⁢(Ω,ℝn){W^{1,p}(\Omega,\mathbb{R}^{n})}, for some p⩾1{p\geqslant 1}, to a mapping f . We show that J⁢(x,f)⩾0{J(x,f)\geqslant 0} a.e. in Ω. Generalizations to higher dimensions are also given.

We prove some existence results for the following Schrödinger–Maxwell system of elliptic equations: {-÷(M(x)∇u)+Aφ|u|r-2u=f,u∈W01,2⁢(Ω),-÷(M(x)∇φ)=|u|r,φ∈W01,2⁢(Ω).\left\{\begin{aligned} &\displaystyle{-}\div(M(x)\nabla u)+A\varphi|u|^{r-2}u=% f,&&\displaystyle u\in W_{0}^{1,2}(\Omega),\\ &\displaystyle{-}\div(M(x)\nabla\varphi)=|u|^{r},&&\displaystyle\varphi\in W_{% 0}^{1,2}(\Omega).\end{aligned}\right. In particular, we prove the existence of a finite energy solution (u,φ){(u,\varphi)} if r>2*{r>2^{*}} and f does not belong to the “dual space” L2⁢NN+2⁢(Ω){L^{\frac{2N}{N+2}}(\Omega)}.

We illustrate a Bellman function technique in finding the modulus of uniform convexity of Lp{L^{p}} spaces.

In this paper, the so-called relative ∞{\infty}-capacity is introduced and investigated in a close connection to the viscosity solution of the ∞{\infty}-Laplace equation. We not only show that the relative ∞{\infty}-capacity equals the limit of the p -th root of the relative p -capacity as p→∞{p\to\infty} and hence has a simple geometric characterization in terms of the Euclidean distance, but also establish several basic properties for the relative ∞{\infty}-capacity. Consequently, we apply the relative ∞{\infty}-capacity to the embedding theory of the ∞{\infty}-Sobolev space. More geometrically, we affinize the relative ∞{\infty}-capacity and its fundamental features as much as possible.