Band 15, Heft 3

We show how to derive (variants of) Michell truss theory in two and three dimensions rigorously as the vanishing weight limit of optimal design problems in linear elasticity in the sense of Γ-convergence. We improve our results from [H. Olbermann, Michell trusses in two dimensions as a Γ-limit of optimal design problems in linear elasticity, Calc. Var. Partial Differential Equations 56 2017, 6, Article ID 166] in that our treatment here includes the three-dimensional case and that we allow for more general boundary conditions and applied forces.

Building upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in ℝ n + 1 {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null ℋ n - 1 {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results: (i) local finiteness of the ( n - 1 ) {(n-1)} -dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) ℋ n - 1 {{\mathcal{H}}^{n-1}} -rectifiability of the free boundary, (iii) classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most ( n - 2 ) {(n-2)} in the free boundary. Instead, if φ ∈ C k + 1 ⁢ ( ℝ n ) {\varphi\in C^{k+1}(\mathbb{R}^{n})} , k ≥ 2 {k\geq 2} , similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than k + 1 {k+1} .

We prove a homogenization theorem for quadratic convolution energies defined in perforated domains. The corresponding limit is a Dirichlet-type quadratic energy, whose integrand is defined by a non-local cell-problem formula. The proof relies on an extension theorem from perforated domains belonging to a wide class containing compact periodic perforations.

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as t → ∞ {t\to\infty} .

We consider functionals of the form ℱ ⁢ ( v , Ω ) = ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) ⁢ 𝑑 x , \mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,dx, with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W 1 , q {W^{1,q}} . We prove a higher differentiability result for the minimizers. We also infer a Lipschitz regularity result of minimizers if q > n {q>n} , and a result of higher integrability for the gradient if q = n {q=n} . The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

We study the comparison principle for non-negative solutions of the equation ∂ ⁡ ( | v | p - 2 ⁢ v ) ∂ ⁡ t = div ⁡ ( | ∇ ⁡ v | p - 2 ⁢ ∇ ⁡ v ) , 1 < p < ∞ . \frac{\partial(|v|^{p-2}v)}{\partial t}=\operatorname{div}(|\nabla v|^{p-2}% \nabla v),\quad 1<p<\infty. This equation is related to extremals of Poincaré inequalities in Sobolev spaces. We apply our result to obtain pointwise control of the large time behavior of solutions.

In this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} in R n R^{n} or 𝑇 is an optimal transport map for the Monge problem with cost function d ⁢ ( x , y ) d(x,y) , the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} . Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others.

In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation procedure. This makes the theory unclear and quite chaotic to a nonspecialist. The aim of this paper is to fix all the misprints, to follow correct procedures, to exhibit, possibly, the shortest and most elegant proofs and to give a complete and self-contained overview of the theory.

The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model n -dimensional hyperbolic space, complementing the results of Borisov and Freitas (2017), Hurtado, Markvorsen and Palmer (2016) and Savo (2008); in odd dimensions, such eigenvalues appear as roots of an inductively constructed transcendental equation. We then give a synthetic proof of Cheng’s sharp eigenvalue comparison theorem in metric measure spaces satisfying a Bishop–Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.

We consider the parabolic polyharmonic diffusion and the L 2 {L^{2}} -gradient flow for the square integral of the m -th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L 2 {L^{2}} , then the evolving curve converges exponentially in the C ∞ {C^{\infty}} topology to a straight horizontal line segment. The same behaviour is shown for the L 2 {L^{2}} -gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m .

In this paper we study the ( BV , L p ) {(\mathrm{BV},L^{p})} -decomposition, p = 1 , 2 {p=1,2} , of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case p = 1 {p=1} we also study the associated geometric problem and the thresholding parameters describing the behavior of its solutions.

We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler–Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous Lagrangians of compact range, this measure solves the Euler–Lagrange equations. Furthermore, we prove that the constructed measure is a minimizer under variations of compact support. Under additional assumptions, it is proven that this measure is a minimizer under variations of finite volume. We finally extend our results to continuous Lagrangians decaying in entropy.

We introduce a notion of geodesic curvature k ζ k_{\zeta} for a smooth horizontal curve 𝜁 in a three-dimensional contact sub-Riemannian manifold, measuring how much a horizontal curve is far from being a geodesic. We show that the geodesic curvature appears as the first corrective term in the Taylor expansion of the sub-Riemannian distance between two points on a unit speed horizontal curve d SR 2 ⁢ ( ζ ⁢ ( t ) , ζ ⁢ ( t + ε ) ) = ε 2 - k ζ 2 ⁢ ( t ) 720 ⁢ ε 6 + o ⁢ ( ε 6 ) . d_{\mathrm{SR}}^{2}(\zeta(t),\zeta(t+\varepsilon))=\varepsilon^{2}-\frac{k_{\zeta}^{2}(t)}{720}\varepsilon^{6}+o(\varepsilon^{6}). The sub-Riemannian distance is not smooth on the diagonal; hence the result contains the existence of such an asymptotics. This can be seen as a higher-order differentiability property of the sub-Riemannian distance along smooth horizontal curves. It generalizes the previously known results on the Heisenberg group.

The paper deals with the existence of nontrivial solutions for ( p , Q ) (p,Q) equations in the Heisenberg group H n \mathbb{H}^{n} with critical exponential growth at infinity and a singular behavior at the origin. The main features and novelty of the paper are the above generality on the right-hand side of the equation, the ( p , Q ) (p,Q) growth of the elliptic operator and the fact that the equation is studied in the entire Heisenberg group.