Band 18, Heft 4

We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set Ω and a thin layer Σ locally described by ε ⁢ h {\varepsilon h} , where h is a positive function defined on the boundary ∂ ⁡ Ω {\partial\Omega} , and ε is the ellipticity constant of the differential operator in the thin layer Σ. We study the problem in the limit for ε going to zero and prove a first-order asymptotic development by Γ-convergence of the associated energy functional.

In this paper we study the degenerate parabolic p -Laplacian, ∂ t ⁡ u - v - 1 ⁢ div ⁡ ( | Q ⁢ ∇ ⁡ u | p - 2 ⁢ Q ⁢ ∇ ⁡ u ) = 0 {\partial_{t}u-v^{-1}\operatorname{div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=0} , where the degeneracy is controlled by a matrix Q and a weight v . With mild integrability assumptions on Q and v , we prove the existence and uniqueness of solutions on any interval [ 0 , T ] {[0,T]} . If we further assume the existence of a degenerate Sobolev inequality with gain, the degeneracy again controlled by v and Q , then we can prove both finite time extinction and ultracontractive bounds. Moreover, we show that there is equivalence between the existence of ultracontractive bounds and the weighted Sobolev inequality.

We refine the iterated blow-up techniques. This technique, combined with a rigidity result and a specific choice of the kernel projection in the Poincaré inequality, might be employed to completely linearize blow-ups along at least one sequence. We show how to implement such argument by applying it to derive affine blow-up limits for BD {\mathrm{BD}} and BV {\mathrm{BV}} functions around Cantor points. In doing so we identify a specific subset of points – called totally singular points having blow-ups with completely singular gradient measure D ⁢ p = D s ⁢ p {Dp=D^{s}p} , ℰ ⁢ p = ℰ s ⁢ p {{\mathcal{E}}p={\mathcal{E}}^{s}p} – at which such linearization fails.

In this article we study the H 1 {H^{1}} -gradient flow for the energy E ⁢ [ X ] {E[X]} given by the quotient of the Dirichlet energy and the signed enclosed area of an H 1 {H^{1}} map X : 𝕊 → ℝ 2 {X:\mathbb{S}\rightarrow\mathbb{R}^{2}} . We prove that solutions with initially positive signed enclosed area exist eternally, and converge as t → ∞ {t\rightarrow\infty} to a (possibly multiply-covered) circle. In this way we recover an improved parametrised isoperimetric inequality for H 1 {H^{1}} maps.

We consider a free-boundary and free-discontinuity energy connecting phase separation and fracture in an elastic material. The energy excludes the contribution of phase boundaries in the cracked region, providing a heuristic approximation of the interfacial energy in the current material configuration. Our primary result shows that the sharp energy may be recovered via Γ-convergence from a modified Cahn–Hilliard energy coupled with an Ambrosio–Tortorelli-type approximation of the (linear) elastic and fracture energy.

We compute an upper bound for the value of the L 1 {L^{1}} -relaxed area of the graph of the vortex map u : B l ⁢ ( 0 ) ⊂ ℝ 2 → ℝ 2 {u:{\rm B}_{l}(0)\subset\mathbb{R}^{2}\to\mathbb{R}^{2}} , u ⁢ ( x ) := x | x | {u(x):=\frac{x}{|x|}} , x ≠ 0 {x\neq 0} , for all values of l > 0 {l>0} . Together with a previously proven lower bound, this upper bound turns out to be optimal. Interestingly, for the radius l in a certain range, in particular l not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved.

We prove that Palais–Smale sequences for Liouville-type functionals on closed surfaces are precompact whenever they satisfy a bound on their Morse index. As a byproduct, we obtain a new proof of existence of solutions for Liouville-type mean-field equations in a supercritical regime. Moreover, we also discuss an extension of this result to the case of singular Liouville equations.

In this paper we provide a characterization of sets with finite local and non-local perimeter via a Γ-convergence result. As an application, we give a short proof of the isoperimetric inequality, both in the local and in the non-local cases.

In this paper, we investigate properties of metric projections onto specific closed and geodesically convex proper subsets of Wasserstein spaces ( P p ⁢ ( R d ) , W p ) (\mathcal{P}_{p}(\mathbf{R}^{d}),W_{p}) . When d = 1 d=1 , as ( P 2 ⁢ ( R ) , W 2 ) (\mathcal{P}_{2}(\mathbf{R}),W_{2}) is isometrically isomorphic to a flat space with a Hilbertian structure, the corresponding projection operators are expected to be nonexpansive. We give a direct proof of this fact, relying on intrinsic analysis, which also implies nonexpansiveness in certain special cases in higher dimensions. When d > 1 d>1 , we show the failure of this property in two regimes: when p > 1 p>1 is either small enough or large enough. Finally, we prove some positive curvature properties of Wasserstein spaces ( P p ⁢ ( R d ) , W p ) (\mathcal{P}_{p}(\mathbf{R}^{d}),W_{p}) when d ≥ 2 d\geq 2 and p ∈ ( 1 , + ∞ ) p\in(1,+\infty) are arbitrary: we show that Wasserstein spaces are nowhere locally Busemann NPC spaces, and they nowhere locally satisfy the so-called projection criterion . As a corollary of the former, they have nonnegative upper Alexandrov curvature , in a precise sense that we define here. In our analysis, a particular subset of probability measures having densities uniformly bounded above by a given constant plays a special role.

We present a minimization problem with a horizontal divergence-type constraint in the Heisenberg group. Our study explores its dual formulation and examines its relationship with the congested optimal transport problem, for 1 < p < + ∞ {1<p<+\infty} , as well as the Monge–Kantorovich problem, in the limit case p = 1 {p=1} .

We establish universality of the renormalised energy for mappings from a planar domain to a compact manifold, by approximating subquadratic polar convex functionals of the form ∫ Ω f ⁢ ( | D ⁢ u | ) ⁢ d x {\int_{\Omega}f(|\mathrm{D}u|)\,\mathrm{d}x} . The analysis relies on the condition that the vortex map x | x | {\frac{x}{\lvert x\rvert}} has finite energy and that t ↦ f ⁢ ( t ) {t\mapsto f(\sqrt{t})} is concave. We derive the leading order asymptotics and provide a detailed description of the convergence of W 1 , 1 {\mathrm{W}^{1,1}} -almost minimisers, leading to a characterization of second-order asymptotics. At the core of the method, we prove a ball merging construction (following Jerrard and Sandier’s approach) for a general class of convex integrands. We therefore generalize the approximation by p -harmonic mappings when p ↗ 2 {p\nearrow 2} and can also cover linearly growing functionals, including those of area type.

We consider a variational model for heterogeneous phase separation, based on a diffuse interface energy with moving wells. Our main result identifies the asymptotic behavior of the first variation of the phase field energies as the width of the diffuse interface vanishes. This convergence result allows us to deduce a Gibbs–Thomson relation for heterogeneous surface tensions. Proceeding from this information, we prove that (weak) solutions of the Allen–Cahn equation with space dependent potential converge to a BV solution of weighted mean curvature flow, under an energy convergence hypothesis. Additionally, relying on the relative energy technique, we establish a weak-strong uniqueness principle for solutions of weighted mean curvature flow.

Harnack inequality and consequent regularity is proved for weak solutions of certain degenerate elliptic equations.

In this paper we prove a strong two-scale approximation result for sphere-valued maps in the space L 2 ⁢ ( Ω ; W 0 1 , 2 ⁢ ( Q 0 ; 𝕊 2 ) ) {L^{2}(\Omega;W^{1,2}_{0}(Q_{0};\mathbb{S}^{2}))} , where Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} is an open domain and Q 0 ⊂ Q {Q_{0}\subset Q} an open subset of the unit cube Q = ( 0 , 1 ) 3 {Q=(0,1)^{3}} . The proof relies on a generalization of the seminal argument by F. Bethuel and X. M. Zheng to the two-scale setting. We then present an application to a variational problem in high-contrast micromagnetics.

In this paper, we establish maximal function estimates, Lebesgue differentiation theory, Calderón–Zygmund decompositions, and John–Nirenberg inequalities for translation invariant Hausdorff contents. We further identify a key structural component of these results – a packing condition satisfied by these Hausdorff contents which compensates for the non-linearity of the capacitary integrals. We prove that for any outer capacity, this packing condition is satisfied if and only if the capacity is equivalent to its induced Hausdorff content. Finally, we use this equivalence to extend the preceding theory to general outer capacities which are assumed to satisfy this packing condition.