Volume 18, Issue 2

In this work, a generalized Hopf’s lemma and a global boundary Harnack inequality are proved for solutions to fractional p -Laplacian equations. Then the isolation of the first ( s , p ) {(s,p)} -eigenvalue is shown in bounded open sets satisfying the Wiener criterion.

We prove the existence of a distributional solution to a quasilinear system of degenerate equations when the datum has low integrability. The main assumption asks the off-diagonal coefficients to have support in a crossed staircase set.

Given is a bounded domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , and a vector-valued function defined on ∂ ⁡ Ω {\partial\Omega} (depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing ∂ ⁡ Ω {\partial\Omega} with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function 𝐮 = ( u 1 , … , u m ) {\mathbf{u}=(u^{1},\dots,u^{m})} ( m ≥ 1 {m\geq 1} ) that represents the temperature within Ω and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, the Bernoulli free boundary problem.

We consider a system of PDEs that model a viscoelastic fluid with damping mechanism. In ℝ 3 {\mathbb{R}^{3}} , we construct some new local energy bounds that enable us to improve several ϵ-regularity criteria for the Caffarelli–Kohn–Nirenberg theorem for weak solutions of this system.

Given a bounded open connected set Ω ⊂ ℝ 2 {\Omega\subset\mathbb{R}^{2}} with Lipschitz boundary, we consider the class of piecewise constant maps u taking three fixed values α , β , γ ∈ ℝ 2 {\alpha,\beta,\gamma\in\mathbb{R}^{2}} , vertices of an equilateral triangle; for any u in this class, using a weak notion of Jacobian determinant valid for BV functions, we give a precise description of Det ⁡ ( ∇ ⁡ u ) {\operatorname{Det}(\nabla\kern 0.569055ptu)} and show that the relaxed graph area of u is bounded from above by a quantity related to the flat norm of Det ⁡ ( ∇ ⁡ u ) {\operatorname{Det}(\nabla\kern 0.569055ptu)} . The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions.

In a geometrically non-linear Cosserat model for micro-polar elastic solids, we prove that critical points of the Cosserat energy functional with an arbitrary large (finite) number of singularities do exist, whereas Cosserat energy minimizers are known to be locally Hölder continuous. To reach that goal, we first develop a technique to insert dipole pairs of singularities into smooth maps while controlling the amount of Cosserat energy needed to do so. We then use this method to force an arbitrary number of singular points into (weak) Cosserat-elastic solids by prescribing smooth boundary data. The boundary data themselves are given in such a way, that they contain no topological obstruction to regularity. Throughout this paper, we often exploit connections between harmonic maps and Cosserat-elastic solids, so that we are able to adapt and incorporate ideas of R. Hardt and F.-H. Lin for harmonic maps with singularities, as well as of F. Béthuel for dipole pairs of singularities.

We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the distance to a geodesic sphere of an embedded hypersurface with integral norms of the traceless Hessian operator of a level set function for the open set bounded by the hypersurface. As application, we give a unified treatment of many old and new stability problems arising in geometry and analysis. Those problems ask for spherical closeness of a hypersurface, given a geometric constraint. Examples include stability in Alexandroff’s soap bubble theorem in space forms, Serrin’s overdetermined problem, a Steklov problem involving the bi-Laplace operator and non-convex Alexandroff–Fenchel inequalities.

We study the regularity of the free boundary in the fully nonlinear parabolic thin obstacle problem. Under the assumption of time semiconvexity, our main result establishes that the free boundary is a C 1 C^{1} graph in x near any regular free boundary point.

We extend the Calderón–Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations ℒ A s ⁢ u = f {\mathcal{L}^{s}_{A}u=f} , where the operator ℒ A s {\mathcal{L}^{s}_{A}} is formally given by ℒ A s ⁢ u = ∫ ℝ n A ⁢ ( x , y ) | x - y | n + 2 ⁢ s ⁢ ( x - y ) ⊗ ( x - y ) | x - y | 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ 𝑑 y . \mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-% y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For 0 < s < 1 {0<s<1} and A : ℝ n × ℝ n → ℝ {A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}} taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier–Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if A ⁢ ( ⋅ , y ) {A(\,\cdot\,,y)} is uniformly Holder continuous and inf x ∈ ℝ n ⁡ A ⁢ ( x , x ) > 0 {\inf_{x\in\mathbb{R}^{n}}A(x,x)>0} , then for f ∈ L loc p {f\in L^{p}_{\rm loc}} , for p ≥ 2 {p\geq 2} , the solution vector u ∈ H loc 2 ⁢ s - δ , p {u\in H^{2s-\delta,p}_{\rm loc}} for some δ ∈ ( 0 , s ) {\delta\in(0,s)} .

We continue our study, started in [A. Češík, G. Gravina and M. Kampschulte, Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision, Calc. Var. Partial Differential Equations 63 2024, 2, Paper No. 55], of (self-)collisions of viscoelastic solids in an inertial regime. We show existence of weak solutions with a corresponding contact force measure in the case of solids with only Lipschitz-regular boundaries. This necessitates a careful study of different concepts of tangent and normal cones and the role these play both in the proofs and in the formulation of the problem itself. Consistent with our previous approach, we study contact without resorting to penalization, i.e., by only relying on a strict non-interpenetration condition. Additionally, we improve the strategies of our previous proof, eliminating the need for regularization terms across all levels of approximation.

We propose a discrete lattice model of the energy of dislocations in three-dimensional crystals which properly accounts for lattice symmetry and geometry, arbitrary harmonic interatomic interactions, elastic deformations and discrete crystallographic slip on the full complement of slip systems of the crystal class. Under the assumption of diluteness, we show that the discrete energy converges, in the sense of Γ-convergence, to a line-tension energy defined on Volterra line dislocations, regarded as integral vector-valued currents supported on rectifiable curves. Remarkably, the line-tension limit is of the same form as that derived from semidiscrete models of linear elastic dislocations based on a core cutoff regularization. In particular, the line-tension energy follows from a cell relaxation and differs from the classical ansatz, which is quadratic in the Burgers vector.

We consider a model to describe stable configurations in epitaxial growth of crystals in the two-dimensional case, and in the regime of linearized elasticity. The novelty is that the model also takes into consideration the adatom density on the surface of the film. These are behind the main mechanisms of crystal growth and formation of islands (or quantum dots). The main result of the paper is the integral representation of the relaxed energy.

It is well known that the Willmore flow of closed spherical immersions exists globally in time and converges if the initial datum has Willmore energy below 8 ⁢ π 8\pi , exactly the Li–Yau energy threshold below which all closed immersions are embedded. Extending the Li–Yau inequality for closed surfaces via Simon’s monotonicity formula also for surfaces with boundary, given Dirichlet boundary conditions, one obtains an energy threshold C LY C_{\mathrm{LY}} below which surfaces with this boundary are embedded. By a slight modification, one obtains a threshold C LY rot C_{\mathrm{LY}}^{\mathrm{rot}} below which surfaces of revolution satisfying the boundary data have no self-intersections on the rotation axis. With a new argument, using this modified Li–Yau inequality and tools from geometric measure theory, we show that the Willmore flow with Dirichlet boundary data starting in cylindrical surfaces of revolution exists globally in time if the energy of the initial datum is below C LY rot C_{\mathrm{LY}}^{\mathrm{rot}} . Moreover, given Dirichlet boundary data, we also obtain the existence of a Willmore minimizer in the class of cylindrical surfaces of revolution if the corresponding infimum lies below C LY rot C_{\mathrm{LY}}^{\mathrm{rot}} , which improves previous results for the stationary problem.

Starting from a model of nonlinear magnetoelasticity where magnetization is defined in the Eulerian configuration while elastic deformation is in the Lagrangian one, we rigorously derive a linearized model that coincides with the standard one that already appeared in the literature and where the zero-stress strain is quadratic in the magnetization. The relation of the nonlinear and linear model is stated in terms of the Γ-convergence and convergence of minimizers.

We obtain a weak formulation of the stationarity condition for the half Dirichlet energy, which can be expressed in terms of a fractional quantity, related to the trace of the Hopf differential of the harmonic extension of the original map. As an application we show that conformal harmonic maps from the disc are precisely the harmonic extensions of stationary points of the half Dirichlet energy on the circle. We also derive a Noether theorem and a Pohozaev identity for stationary points of the half Dirichlet energy.