Band 17, Heft 1

Let Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} be a C 1 {C^{1}} smooth compact domain. Furthermore, let F : Ω × ℝ n ⁢ N → ℝ {F:\Omega\times\mathbb{R}^{nN}\to\mathbb{R}} , F ⁢ ( x , p ) {F(x,p)} , be C 0 {C^{0}} , differentiable with respect to p , and with F p := D p ⁢ F {F_{p}:=D_{p}F} continuous on Ω × ℝ n ⁢ N {\Omega\times\mathbb{R}^{nN}} and F strictly convex in p . Consider an n ⁢ N × n ⁢ N {nN\times nN} matrix A = ( A α ⁢ β i ⁢ j ) ∈ C 0 ⁢ ( Ω ) {A=(A^{{ij}}_{\alpha\beta})\in C^{0}(\Omega)} satisfying (0.1) A α ⁢ β i ⁢ j ⁢ ( x ) ⁢ ξ α i ⁢ ξ β j = A β ⁢ α j ⁢ i ⁢ ( x ) ⁢ ξ α i ⁢ ξ β j ≥ λ ⁢ | ξ | 2 , λ > 0 . A^{ij}_{\alpha\beta}(x)\xi^{i}_{\alpha}\xi^{j}_{\beta}=A^{ji}_{\beta\alpha}(x)% \xi^{i}_{\alpha}\xi^{j}_{\beta}\geq\lambda\lvert\xi\rvert^{2},\quad\lambda>0. Suppose that (0.2) lim | p | → ∞ ⁡ 1 | p | ⁢ ( D p ⁢ F ⁢ ( x , p ) - A ⁢ ( x ) ⁢ p ) = 0 , \displaystyle\lim_{\lvert p\rvert\to\infty}\frac{1}{\lvert p\rvert}(D_{p}F(x,p% )-A(x)p)=0, (0.3) - C 0 + c 0 ⁢ | p | 2 ≤ F ⁢ ( x , p ) ≤ C 0 ⁢ ( 1 + | p | 2 ) , \displaystyle{-}C_{0}+c_{0}\lvert p\rvert^{2}\leq F(x,p)\leq C_{0}(1+\lvert p% \rvert^{2}), (0.4) | F p ⁢ ( x , p ) - F p ⁢ ( x , q ) | ≤ C 0 ⁢ | p - q | , \displaystyle\lvert F_{p}(x,p)-F_{p}(x,q)\rvert\leq C_{0}\lvert p-q\rvert, (0.5) 〈 F p ⁢ ( x , p ) - F p ⁢ ( x , q ) , p - q 〉 ≥ c 0 ⁢ | p - q | 2 \displaystyle\langle F_{p}(x,p)-F_{p}(x,q),p-q\rangle\geq c_{0}\lvert p-q% \rvert^{2} uniformly in x and with positive constants c 0 {c_{0}} and C 0 {C_{0}} . Consider the functional (0.6) J ⁢ ( u ) := ∫ Ω F ⁢ ( x , D ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x + ∫ Ω G ⁢ ( x , u ) ⁢ 𝑑 x , J(u):=\int_{\Omega}F(x,Du(x))\,dx+\int_{\Omega}G(x,u)\,dx, where G ⁢ ( x , ⋅ ) ∈ C 1 ⁢ ( ℝ N ) {G(x,\cdot\,)\in C^{1}(\mathbb{R}^{N})} for each x ∈ Ω {x\in\Omega} , G ⁢ ( ⋅ , u ) {G(\,\cdot\,,u)} is measurable for each u ∈ ℝ N {u\in\mathbb{R}^{N}} , and (0.7) | G u ⁢ ( x , u ) | ≤ C 0 ⁢ ( 1 + | u | s ) \lvert G_{u}(x,u)\rvert\leq C_{0}(1+\lvert u\rvert^{s}) with s < n + 2 n - 2 {s<\frac{n+2}{n-2}} . Under these conditions, we shall show that if n > 2 {n>2} , then any weak solution u ∈ W 1 , 2 ⁢ ( Ω , ℝ N ) {u\in W^{1,2}(\Omega,\mathbb{R}^{N})} of the Euler equations of J , i.e. ∑ α ∂ ∂ ⁡ x α ⁢ F p α i ⁢ ( x , D ⁢ u ) = G u i ⁢ ( x , u ) , i = 1 , … , N , \sum_{\alpha}\frac{\partial}{\partial x^{\alpha}}F_{p^{i}_{\alpha}}(x,Du)=G_{u% ^{i}}(x,u),\quad i=1,\ldots,N, is Hölder continuous in the interior of Ω and under appropriate boundary conditions also Hölder continuous up to the boundary.

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings.

This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established in [J. Fischer, S. Hensel, T. Laux and T. M. Simon, The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions, preprint 2020, https://arxiv.org/abs/2003.05478], under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows.

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the p -Laplace operator and a general nonlinearity satisfying concavity-type assumptions. This provides an extension of results previously known in the literature only for the torsion and the first eigenvalue equations. In the semilinear case p = 2 {p=2} the results are already new since they include new admissible nonlinearities.

In this article, we prove the various minimality of the product of a 1-codimensional calibrated manifold and a paired calibrated set. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets – Plateau’s problem in the setting of sets. The Almgren minimality was introduced by Almgren to modernize Plateau’s problem. It gives a very good description of local behavior for soap films. The natural question of whether the product of any two Almgren minimal sets is still minimal is still open, although it seems obvious in intuition. We prove the Almgren minimality for the product of two large classes of Almgren minimal sets – the class of 1-codimensional calibrated manifolds and the class of paired calibrated sets. The general idea is to properly combine different topological conditions (separation and spanning) under different homology groups, to set up a reasonable topological condition and prove the minimality for the product under this condition, which will imply the Almgren minimality. A main difficulty comes from the codimension – algebraic coherences such as multiplicity, separation and orientation do not exist anymore for codimensions larger than 1. An unexpectedly useful thing in the present paper is the flow of the calibrations. Its most important role among all is helping us to do the decomposition of a competitor with the help of the first projections along the flows.

We study the Neumann and Dirichlet problems for the total variation flow in doubling metric measure spaces supporting a weak Poincaré inequality. We prove existence and uniqueness of weak solutions and study their asymptotic behavior. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for L 1 {L^{1}} initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss–Green formula.

We prove the absence of the Lavrentiev gap for non-autonomous functionals ℱ ⁢ ( u ) ≔ ∫ Ω f ⁢ ( x , D ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , \mathcal{F}(u)\coloneqq\int_{\Omega}f(x,Du(x))\,dx, where the density f ⁢ ( x , z ) {f(x,z)} is α-Hölder continuous with respect to x ∈ Ω ⊂ ℝ n {x\in\Omega\subset\mathbb{R}^{n}} , it satisfies the ( p , q ) {(p,q)} -growth conditions | z | p ⩽ f ⁢ ( x , z ) ⩽ L ⁢ ( 1 + | z | q ) , \lvert z\rvert^{p}\leqslant f(x,z)\leqslant L(1+\lvert z\rvert^{q}), where 1 < p < q < p ⁢ ( n + α n ) {1<p<q<p(\frac{n+\alpha}{n})} , and it can be approximated from below by suitable densities f k {f_{k}} .

In this paper, when studying the connection between the fractional convexity and the fractional p -Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ⁢ ( x ) {u(x)} is the viscosity sub-solution of the equation if and only if u ⁢ ( x ) {u(x)} is so-called ( α , p ) {(\alpha,p)} -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an ( α , p ) {(\alpha,p)} -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation.

In this paper, we study the existence and interior W 2 , p {W^{2,p}} -regularity of the solution, and the regularity of the free boundary ∂ ⁡ { u > ϕ } {\partial\{u>\phi\}} to the obstacle problem of the porous medium equation, u t = Δ ⁢ u m {u_{t}=\Delta u^{m}} ( m > 1 {m>1} ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ ⁡ { u > ϕ } {\partial\{u>\phi\}} and ∂ ⁡ { u > 0 } {\partial\{u>0\}} , we consider two cases on the initial data which make the free boundary ∂ ⁡ { u > ϕ } {\partial\{u>\phi\}} separate from the free boundary ∂ ⁡ { u > 0 } {\partial\{u>0\}} . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 {C^{1}} -regularity of the free boundary ∂ ⁡ { u > ϕ } {\partial\{u>\phi\}} is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator.

We prove a fractional Pohozaev-type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to the nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.