Regularity Of Minimizers Research Articles

Layer solutions for a one-dimensional nonlocal model of Ginzburg–Landau type

We study a nonlocal model of Ginzburg–Landau type that gives rise to an equation involving a mixture of the Laplacian and half-Laplacian. Our focus is on one-dimensional transition layer profiles that connect the two distinct homogeneous phases. We first introduce a renormalized one-dimensional energy that is free from a logarithmic divergence due to the failure of the Gagliardo norm to be finite on smooth profiles that asymptote to different limits at infinity. We then prove existence, uniqueness, monotonicity and regularity of minimizers in a suitable class. Lastly, we consider the singular limit in which the coefficient in front of the Laplacian vanishes and prove convergence of the obtained minimizer to the solutions of the fractional Allen–Cahn equation.

Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics

Well-known conjectures of Tian predict that the existence of canonical Kähler metrics should be equivalent to various notions of properness of Mabuchi’s K-energy functional. First, we provide counterexamples to Tian’s first conjecture in the presence of continuous automorphisms. Second, we resolve Tian’s second conjecture, confirming the Moser–Trudinger inequality for Fano manifolds. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal replacements for Tian’s first conjecture in the setting of smooth and singular Kähler–Einstein metrics, with or without automorphisms, as well as for Kähler–Ricci solitons. Moreover, we reduce both Tian’s original first conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion.

A higher order TV-type variational problem related to the denoising and inpainting of images

We give a comprehensive survey on a class of higher order variational problems which are motivated by applications in mathematical imaging. The overall aim of this note is to investigate if and in which manner results from the first author’s previous work on variants of the TV-regularization model (see e.g. Bildhauer and Fuchs (2012, 2014, 2015) and Fuchs and Tietz (2015)) can be extended to functionals which involve higher derivatives. This seems to be not only of theoretical interest, but also relevant to applications since higher order TV-denoising appears to maintain the advantages of the classical model as introduced by Rudin, Osher and Fatemi in (Rudin et al., 1992) while avoiding the unpleasant “staircasing” effect (see e.g. Bredies et al. (2010) or Lysaker et al. (2003)). Our paper features results concerning generalized solutions in spaces of functions of higher order bounded variation, dual solutions as well as partial regularity of minimizers.

Sobolev and Lipschitz regularity for local minimizers of widely degenerate anisotropic functionals

We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.

Solutions to the Reifenberg Plateau problem with cohomological spanning conditions

We prove existence and regularity of minimizers for Holder densities over general surfaces of arbitrary dimension and codimension in \( \mathbb {R}^n \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg’s Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams; in particular we generalize a particular type of minimizing sequence used by Reifenberg (whose limits have nice properties, including lower bounds on lower density and finite Hausdorff measure), prove such minimizing sequences exist, and develop cohomological spanning conditions which generalize spanning conditions using linking numbers and have better properties than homological spanning conditions. Our cohomology lemmas are dual versions of the homology lemmas in the celebrated appendix by Adams found in Reifenberg’s 1960 paper.

Regularity of minimizers under limit growth conditions

It is well known that an integral of the Calculus of Variations satisfying anisotropic growth conditions may have unbounded minimizers if the growth exponents are too far apart. Under sharp assumptions on the exponents we prove the local boundedness of minimizers of functionals with anisotropic p,q-growth, via the De Giorgi method. As a by-product, regularity of minimizers of some non coercive functionals is obtained by reduction to coercive ones.

Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

Here, we extend the weak KAM and Aubry–Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton–Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi’s weak KAM theorem, and describe the asymptotic limit of the generalized Lax–Oleinik semigroup.

$C^{1,\alpha }$-interior regularity for minimizers of a class of variational problems with linear growth related to image inpainting

$C^{1,\alpha }$-interior regularity for minimizers of a class of variational problems with linear growth related to image inpainting

Regularity of Local Minimizers of the Interaction Energy Via Obstacle Problems

The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems.

Higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients

We study the local regularity of vectorial minimizers of integral functionals with standard p-growth. We assume that the non-homogeneous densities are uniformly convex and have a radial structure, with respect to the gradient variable, only at infinity. Under a W1,n-Sobolev dependence on the spatial variable of the integrand, n being the space dimension, we show that the minimizers have the gradient locally in Lq for every q>p. As a consequence, they are locally α-Hölder continuous for every α<1.

Average-distance problem for parameterized curves

We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite compactly supported measure μ , with \hbox{$\mu\left(\R^d\right)>0$} for p ≥ 1 and λ > 0 we consider the functional \begin{equation} E(\gm) = \int_{\R^d} d(x, \Gamma_\gamma)^p {\rm d}\mu(x) + \lambda \,\text{Length}(\gamma) \end{equation} E ( γ ) = ∫ R d d ( x, Γ γ ) p d μ ( x ) + λ Length ( γ ) where γ : I → ℝ d , I is an interval in ℝ, Γ γ = γ ( I ), and d ( x, Γ γ ) is the distance of x to Γ γ . The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure ℋ 1 , and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures μ supported in two dimensions the minimizing curve is injective if p ≥ 2 or if μ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation.

Regularity for minimizers of integrals with nonstandard growth

We deal with variational integrals ∫Ωf(x,Du(x))dx and we consider a minimizer u:Ω⊂Rn→R among all functions that agree on the boundary ∂Ω with some fixed boundary value u∗. We assume that the boundary datum u∗ makes the density f(x,Du∗(x)) more integrable and we prove that the minimizer u enjoy higher integrability.

Partial regularity for minimizers of singular energy functionals, with application to liquid crystal models

where F is quasiconvex in the gradient variables and the convex function f blows up to infinity at the boundary of a given bounded open set K ⊂ R . As we will explain later in Section 5, this sort of energy functional arises in some recently proposed models in nematic liquid crystal theory. We assume hereafter that U ⊂ R is bounded smooth domain and that K is a bounded, open convex subset of R. Our assumptions are these: (H1) Hypotheses on f : The given function f : R → [0,∞] is nonnegative, convex and smooth on K ⊂ R. We will write f = f(z). We further require that

Nonlocal quantitative isoperimetric inequalities

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the $$t$$ -perimeter, up to multiplicative constants, controls from above that of the $$s$$ -perimeter, with $$s$$ smaller than $$t$$ . To do this we consider a problem of independent interest: we characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $$t$$ -perimeter and the $$s$$ -perimeter. In particular, we show that balls are the unique minimizers if the volume is sufficiently small, depending on $$t-s$$ , while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all $$s,\,t$$ . When $$s=0$$ this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.

RETRACTED ARTICLE: A $${C^{1,\alpha}}$$ partial regularity result for non-autonomous convex integrals with discontinuous coefficients

We establish the \({C^{1,\alpha}}\) partial regularity of vectorial minimizers of non autonomous convex integral functionals of the type $$\mathcal{F}(u;\,\Omega):=\int_{\Omega}f(x, Du)\, dx,$$ with p-growth into the gradient variable. As a novel feature, we allow discontinuous dependence on the x variable, through a suitable Sobolev function. The Holder’s continuity of the gradient of the minimizers is obtained outside a negligible set and this an unavoidable feature in the vectorial setting. Here, the so called singular set has to take into account also of the possible discontinuity of the coefficients.

Partial regularity of minimizers of p (x )-growth functionals with 1 < p (x ) < 2

We prove partial regularity of minimizers u for p ( x ) -energy functionals of the following type: E ( u ) = ∫ Ω ( A i j α β ( x , u ) D α u i D β u j ) p ( x ) / 2 d x , assuming that A i j α β ( x , u ) and p ( x ) are sufficiently smooth and that p ( x ) is subquadratic. We prove that u ∈ C 0 , α ( Ω 0 ) for some α ∈ ( 0 , 1 ) and an open set Ω 0 ⊂ Ω with H m - γ 1 ( Ω - Ω 0 ) = 0 , where H s denotes the s-dimensional Hausdorff measure and γ 1 = inf Ω p ( x ) .

Local Boundedness of Minimizers with Limit Growth Conditions

The energy integral of the calculus of variations, which we consider in this paper, has a limit behavior when the maximum exponent $$q$$q, in the growth estimate of the integrand, reaches a threshold. In fact, if $$q$$q is larger than this threshold, counterexamples to the local boundedness and regularity of minimizers are known. In this paper, we prove the local boundedness of minimizers (and also of quasi-minimizers) under this stated limit condition. Some other general and limit growth conditions are also considered.

More counterexamples to regularity for minimizers of the average-distance problem

Abstract The average-distance problem, in the penalized formulation, involves minimizing (1) E μ λ(Σ) := ∫ℝ d d(x,Σ)dμ(x) + λℋ 1 (Σ), among path-wise connected, closed sets Σ with finite ℋ 1 -measure, where d ≥ 2, μ is a given measure, λ is a given parameter and d(x,Σ) := inf y∈Σ|x - y|. The average-distance problem can be also considered among compact, convex sets with perimeter and/or volume penalization, i.e. minimizing (2) ℰ(μ,λ1,λ2)(·):=∫ℝ d d(x,·)dμ(x) + λ1Per(·) + λ2Vol(·), where μ is a given measure, λ1,λ2 ≥ 0 are given parameters with λ1 + λ2 &gt; 0, and the unknown varies among compact, convex sets. Very little is known about the regularity of minimizers of (2). In particular it is unclear if minimizers of (2) are in general C 1 regular. The aim of this paper is twofold: first, we provide in ℝ2 a second approach in constructing minimizers of (1) which are not C 1 regular; then, using the same technique, we provide an example of minimizer of (2) whose border is not C 1 regular, under perimeter penalization only.

Regularity of minimizers of autonomous convex variational integrals

We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,\Omega) = \int_{\Omega} /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon \Omega \subset {\mathbb R}^n \to {\mathbb R}^N$ satisfying a Dirichlet boundary condition. The integrands $F$ are assumed to be autonomous, convex and of $(p,q)$ growth, but are otherwise not subjected to any further structure conditions, and we consider exponents in the range $1<p \leq q < p^{\ast}$, where $p^{\ast}$ denotes the Sobolev conjugate exponent of $p$.

Boundary regularity of minimizers of p(x)-energy functionals

The paper is devoted to the study of the regularity on the boundary ∂Ω of a bounded open set Ω⊂Rm for minimizers u for p(x)-energy functionals of the following typeE(u;Ω):=∫Ω(gαβ(x)Gij(u)Dαui(x)Dβuj(x))p(x)/2dx where (gαβ(x)) and (Gij(u)) are symmetric positive definite matrices whose entries are continuous functions and p(x)≥2 is a continuous function. The authors prove that such minimizers u have no singular points on the boundary.