Volume 11, Issue 3

We study a geometric analysis and local regularity for the evolution of p{{p}}-harmonic maps, called p{{p}}-harmonic map heat flows. Our main result is to establish a criterion for a uniform local regularity estimate for regular p{{p}}-harmonic map heat flows, devising some new monotonicity-type formulas of a local scaled energy. The regularity criterion obtained is almost optimal, comparing with that of the corresponding stationary case. As application we show a compactness of regular p{{p}}-harmonic map heat flows with energy bound.

We consider functionals of the form J⁢(u)=12⁢∫ℝN|∇⁡u|2-∫ℝNb⁢(x)⁢G⁢(u)J(u)=\frac{1}{2}\int_{{\mathbb{R}}^{N}}|\nabla u|^{2}-\int_{{\mathbb{R}}^{N}}b% (x)G(u) on a C1{C^{1}}-submanifold M of 𝒟1,2⁢(ℝN){\mathcal{D}^{1,2}({\mathbb{R}}^{N})}, N≥3{N\geq 3}, where G is the primitive of some “zero-mass” nonlinearity g (i.e., g′⁢(0)=0{g^{\prime}(0)=0}), and the weight function b:ℝN→ℝ{b:{\mathbb{R}}^{N}\to{\mathbb{R}}} is merely supposed to belong to L1⁢(ℝN)∩L2*2*-p⁢(ℝN){L^{1}({\mathbb{R}}^{N})\cap L^{\frac{2^{*}}{2^{*}-p}}({\mathbb{R}}^{N})} for some 2<p<2*{2<p<2^{*}}, and to possess a certain decay behavior. Let V be the subspace of 𝒟1,2⁢(ℝN){\mathcal{D}^{1,2}({\mathbb{R}}^{N})} given by V:={v∈𝒟1,2⁢(ℝN):v∈C⁢(ℝN)⁢ with ⁢supx∈ℝN⁡(1+|x|N-2)⁢|v⁢(x)|<∞}{V:=\{v\in\mathcal{D}^{1,2}({\mathbb{R}}^{N}):v\in C({\mathbb{R}}^{N})\mbox{ % with }\sup_{x\in{\mathbb{R}}^{N}}(1+|x|^{N-2})|v(x)|<\infty\}}. We prove that a local minimizer of the constrained functional J|M{J|_{M}} with respect to the V -topology must be a local minimizer with respect to the “bigger” 𝒟1,2⁢(ℝN){\mathcal{D}^{1,2}({\mathbb{R}}^{N})}-topology. This result allows us to prove the existence of multiple nontrivial solutions of the zero-mass equation -Δ⁢u=b⁢(x)⁢g⁢(u){-\Delta u=b(x)g(u)} in ℝN{{\mathbb{R}}^{N}}, where g:R→ℝ{g:R\to{\mathbb{R}}} is a subcritical nonlinearity, which is superlinear at zero and at ∞{\infty}.

We prove existence and regularity results for weak solutions of non-linear elliptic systems with non-variational structure satisfying (p,q){(p,q)}-growth conditions. In particular, we are able to prove higher differentiability results under a dimension-free gap between p and q .

In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular, we here establish an L1{L^{1}} Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler–Lagrange equations obtained as conditions of minimality. In addition, we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.

In this paper, we prove that every equivalence class in the quotient group of integral 1-currents modulo p in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for m -dimensional integral currents modulo p implies that the family of (m-1){(m-1)}-dimensional flat chains of the form pT , with T a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for 0-dimensional flat chains, and, using a proposition from The structure of minimizing hypersurfaces mod 4 by Brian White, also for flat chains of codimension 1.

In this paper, we give a remark on the two-phase obstacle-type problem for the p -Laplacian when 1<p<2{1<p<2} and p is close to 2. We prove that the free boundary where the gradient vanishes has locally finite (n-1){(n-1)}-Hausdorff measure.