Differentiability and higher integrability results for local minimizers of splitting-type variational integrals in 2D with applications to nonlinear Hencky-materials

We prove higher integrability and differentiability results for local minimizers u: ${\mathbb {R}^2\supset\Omega\to\mathbb {R}^M}$ , M ≥ 1, of the splitting-type energy ${\int_{\Omega}[h_1(|\partial_1 u|)+h_2(|\partial_2 u|)]\,{\rm d}x}$ . Here h ₁, h ₂ are rather general N-functions and no relation between h ₁ and h ₂ is required. The methods also apply to local minimizers u: ${\mathbb {R}^2\supset\Omega \to \mathbb {R}^2}$ of the functional ${\int_{\Omega}[h_1(|{\rm div}\,{\rm u}|)+h_2(|\varepsilon^D(u)|)]\,{\rm d}x}$ so that we can include some variants of so-called nonlinear Hencky-materials. Further extensions concern non-autonomous problems.     

Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

In a simply connected two dimensional domain Ω, we consider Ginzburg-Landau minimizers u with zero degree Dirichlet boundary condition ${g \in H^{1/2}(\partial \Omega; \mathbb{S}^1)}$ . We prove uniqueness of u whenever either the energy or the Ginzburg-Landau parameter are small. This generalizes a result of Ye and Zhou requiring smoothness of g. We also obtain uniqueness when Ω is multiply connected and the degrees of the vortexless minimizer u are prescribed on the components of the boundary, generalizing a result of Golovaty and Berlyand for annular domains. The proofs rely on new global estimates connecting the variation of |u| to the Ginzburg-Landau energy of u. These estimates replace the usual global pointwise estimates satisfied by ${\nabla u}$ when g is smooth, and apply to fairly general potentials. In a related direction, we establish new uniqueness results for critical points of the Ginzburg-Landau energy.     

A relaxation of the intrinsic biharmonic energy

The tension field τ(u) of a map u from a domain ${\Omega\subset\mathbb{R}^m}$ into a manifold N is the negative L ²-gradient of the Dirichlet energy. In this paper we study the intrinsic biharmonic energy functional ${T(u) = \int_{\Omega}|\tau(u)|^2}$ . In order to overcome the lack of coercivity of T, we extend it to a larger space. We construct minimizers of the extended functional via the direct method and we study the relation between these minimizers and critical points of T. Our results are restricted to dimensions m ≤ 4.     

Local and global minimizers for a variational energy involving a fractional norm

We study existence, uniqueness, and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int\limits_\Omega W(u)\,{d}x, $$ where ${\|u\|_{H^s(\Omega)}}$ denotes the total contribution from Ω in the H ^s norm of u and W is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space ${\mathbb{R}^n}$ . The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ-convergence and the density estimates for level sets of minimizers.     

Singularity analysis of Ginzburg–Landau energy related to p-wave superconductivity

The following Ginzburg–Landau energy in the absence of a magnetic field $$E_\varepsilon(\psi) = \int\limits_G\left[\frac{1}{2}|\nabla\psi|^2 + \frac{1}{4\varepsilon^2}(1-|\psi|^2)^2\right]{\rm d}x$$ was well studied during recent twenty years. Here, ${G \subset \mathbf{R}^2}$ is a bounded smooth domain, ${\psi}$ is an order parameter, ${\varepsilon >0 }$ . In particular, several global properties including the weighted energy estimation, the concentration compactness properties and the quantization effect of the energy had been established. This paper is concerned with another Ginzburg–Landau type free energy associated with p-wave superconductivity $$E_\varepsilon (\psi, u; G) = \frac{1}{2} \int\limits_G(|\nabla \psi|^2 + |\nabla u|^2 - |\nabla|\psi||^2){\rm d}x + \frac{1}{4\varepsilon^2} \int\limits_G(1-|\psi|^2)^2{\rm d}x.$$ Here, u is also an order parameter. We will prove that those global properties still hold for this more complicated energy functional. Such global properties describe the locations of the regular and the singular domains, and also show the convergence relation between the Ginzburg–Landau minimizers and the harmonic maps.     

Two-phase semilinear free boundary problem with a degenerate phase

We study minimizers of the energy functional $$\int\limits_{D} [|\nabla u|^2+ \lambda(u^+)^p]\,{\rm d}x$$ for ${p\in (0,1)}$ without any sign restriction on the function u. The distinguished feature of the problem is the lack of nondegeneracy in the negative phase. The main result states that in dimension two the free boundaries ${\Gamma^+=\partial\{u>0\}\cap D}$ and ${\Gamma^-=\partial\{u<0\}\cap D}$ are C ^1,?? -regular, provided ${1-\epsilon_0<p<1}$ . The proof is obtained by a careful iteration of the Harnack inequality to obtain a nontrivial growth estimate in the negative phase, compensating for the apriori unknown nondegeneracy.     

Continuity of solutions of a problem in the calculus of variations

We study the problem of minimizing ${\int_{\Omega} L(x,u(x),Du(x))\,{\rm d}x}$ over the functions ${u\in W^{1,p}(\Omega)}$ that assume given boundary values ${\phi}$ on ???. We assume that L(x, u, Du)?=?F(Du)?+?G(x, u) and that F is convex. We prove that if ${\phi}$ is continuous and ?? is convex, then any minimum u is continuous on the closure of ??. When ?? is not convex, the result holds true if F(Du)?=?f(|Du|). Moreover, if ${\phi}$ is Lipschitz continuous, then u is H?lder continuous.     

Gamma-convergence of nonlocal perimeter functionals

Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??).     

Liouville theorems for entire local minimizers of energies defined on the class L log L and for entire solutions of the stationary Prandtl-Eyring fluid model

If ${u: \mathbb{R}^{n} \to \mathbb{R}^{M}}$ locally minimizes the energy with density ${|\nabla u|\ln (1 + |\nabla u|)}$ , then we show that the boundedness of the function u already implies its constancy. The same is true in case n = M = 2 for entire solutions of the equations modelling the stationary flow of a so-called Prandtl-Eyring fluid. Moreover, in the variational setting we will present various extensions of the above mentioned Liouville theorem for entire local minimizers valid in any dimensions n and M.     

Conic volume ratio of the packing cone associate to a convex body

Given a convex body ${K\subset\mathbb{R}^n}$ (n??? 1) which contains o in its interior and ${{\bf u} \in S^{n-1}}$ , we introduce conic volume ratio r(K, u) of K in the direction of u by $$r(K, {\bf u})=\frac{vol(cone(K,{\bf u})\cap B_2^n)}{vol(B_2^n)},$$ where cone(K, u) is the packing cone of K in the direction of u. We prove that if K is an o-symmetric convex body in ${\mathbb{R}^n}$ and r(K, u) is a constant function of u, then K must be a Euclidean ball.     

Percolation in the vacant set of Poisson cylinders

We consider a Poisson point process on the space of lines in ${{\mathbb R}^d}$ , where a multiplicative factor u?>?0 of the intensity measure determines the density of lines. Each line in the process is taken as the axis of a bi-infinite cylinder of radius 1. We investigate percolative properties of the vacant set, defined as the subset of ${{\mathbb R}^d}$ that is not covered by any such cylinder. We show that in dimensions d ≥ 4, there is a critical value ${u_*(d) \in (0,\infty)}$ , such that with probability 1, the vacant set has an unbounded component if u?<?u _*(d), and only bounded components if u?>?u _*(d). For d?=?3, we prove that the vacant set does not percolate for large u and that the vacant set intersected with a two-dimensional subspace of ${{\mathbb R}^d}$ does not even percolate for small u?>?0.     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

On the stability of the L p -norm of the Riemannian curvature tensor

We consider the Riemannian functional \(\mathcal {R}_{p}(g)={\int }_{M}|R(g)|^{p}dv_{g}\) defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M where R(g) and dv _g denote the corresponding Riemannian curvature tensor and volume form and p ∈ (0, ∞). First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for \(\mathcal {R}_{p}\) for certain values of p. Then we conclude that they are strict local minimizers for \(\mathcal {R}_{p}\) for those values of p. Finally generalizing this result we prove that product of space forms of same type and dimension are strict local minimizer for \(\mathcal {R}_{p}\) for certain values of p.     

Democracy functions and optimal embeddings for approximation spaces

We prove optimal embeddings for nonlinear approximation spaces $\mathcal{A}^{\alpha}_q$ , in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for N-term wavelet approximation in L ^p , Orlicz, and Lorentz norms. We also study the ??greedy classes?? ${\mathcal{G}_{q}^{\alpha}}$ introduced by Gribonval and Nielsen, obtaining new counterexamples which show that ${\mathcal{G}_{q}^{\alpha}}\not=\mathcal{A}^{\alpha}_q$ for most non-democratic unconditional bases.     

Degenerate homogeneous parabolic equations associated with the infinity-Laplacian

We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem ${u_t = \Delta_{\infty}^{h} u}$ , where ${\Delta_{\infty}^{h}}$ is the h-homogeneous operator associated with the infinity-Laplacian, ${\Delta_{\infty}^{h} u = |Du|^{h-3} \langle D^{2}uDu, Du \rangle}$ , and h > 1. We also derive the asymptotic behaviour of u for the problem posed in the whole space, and for the Dirichlet problem posed in a bounded domain with zero boundary conditions.     

Regularity for non-autonomous functionals with almost linear growth

We consider non-autonomous functionals ${\mathcal{F}(u; \Omega)=\int_{\Omega}f(x, Du)\ dx}$ , where the density ${f:\Omega\times\mathbb{R}^{nN}\rightarrow\mathbb{R}}$ has almost linear growth, i.e., $$f(x,\xi)\approx |\xi|\log(1+|\xi|).$$ We prove partial C ^1,?? -regularity for minimizers ${u:\mathbb{R}^n\supset\Omega\rightarrow \mathbb{R}^N}$ under the assumption that D _?? f (x, ??) is H?lder continuous with respect to the x-variable. If the x-dependence is C ¹ we can improve this to full regularity provided additional structure conditions are satisfied.     

Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition

We study the Cauchy–Dirichlet problem for the elliptic–parabolic equation $$b(u)_t + {\rm div} F(u) - \Delta u = f$$ in a bounded domain. We do not assume the structure condition $$b(z) = b(\hat z) \Rightarrow F(z) = F(\hat z).$$ Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold (Proc R Soc Edinb 133A(3):477–496, 2003) where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux F, we justify the uniqueness of u (the uniqueness of b(u) is well-known) and prove the continuous dependence in L ¹ for the case of strongly convergent finite energy data. We also prove convergence of the ${\varepsilon}$ -discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux F, we show that the problem admits a maximal and a minimal solution.     

Reachable sets for contact sub-Lorentzian structures on {\mathbb{R}^3} . Application to control affine systems on {\mathbb{R}^3} with a scalar input

In this paper, we investigate the structure of reachable sets for general contact sub-Lorentzian metrics on $ {\mathbb{R}^3} $ . In some particular cases, the presented method leads to explicit formulas for functions describing reachable sets. We also compute the image under exponential mapping and prove that the sub-Lorentzian distance is continuous for the mentioned structures. All presented results concerning reachable sets can be directly applied to generic control affine systems in $ {\mathbb{R}^3} $ with a scalar input u and constraints |u|??????.     

Sharp nonexistence results of prescribed L 2-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations

In this paper, we study the existence of minimizers for $$F(u) = \frac{1}{2} \int_{\mathbb{R}^3} |\nabla u|^{2} {\rm d}x + \frac{1}{4} \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{| u(x)|^2 | u(y)|^2}{| x-y|} {\rm d}x{\rm d}y-\frac{1}{p} \int_{\mathbb{R}^3}|u|^p {\rm d}x$$ on the constraint $$S(c) = \{u \in H^1(\mathbb{R}^3) : \int_{\mathbb{R}^3}|u|^2 {\rm d}x = c\}$$ , where c >  0 is a given parameter. In the range ${p \in [3,\frac{10}{3}]}$ , we explicit a threshold value of c >  0 separating existence and nonexistence of minimizers. We also derive a nonexistence result of critical points of F(u) restricted to S(c) when c >  0 is sufficiently small. Finally, as a by-product of our approaches, we extend some results of Colin et al. (Nonlinearity 23(6):1353–1385, 2010) where a constrained minimization problem, associated with a quasi-linear equation, is considered.