This paper addresses the uniqueness and Lipschitz regularity of solutions for a general minimization problem. We assume that the boundary data satisfy the bounded slope condition to ensure the existence of a minimizer, which is globally Lipschitz continuous. With this regularity, we can prove the existence of a unique solution among functions of bounded variation. Finally, we apply these results to a problem related to Hencky plasticity to prove the continuity of the stress and to investigate the regularity and geometry of the level sets of the minimizer.
In this work, we establish a Serrin-type symmetry result for a class of degenerate elliptic operators that naturally arise in the theory of generalized axially symmetric potentials. The relevant differential operator is L-a = partial derivative(rr)u + a/r partial derivative(r) u + Delta(y)u, where a >= 0 represents the fractional parameter.
This paper investigates the existence of solutions for the following class of coupled systems of partial differential equations with an interface condition{ - Delta u i = lambda i a i ( x ) u i + b i ( x ) | u i | p i - 2 u i in Omega i , partial derivative nu u i = gamma ( u 2 - u 1 ) on Sigma , partial derivative eta u 2 = 0 on Gamma , ight.where Sigma = partial derivative Omega 1, Gamma = partial derivative Omega 2 \ Sigma are sign-changing weight functions. The study is divided into different cases according to the values of the exponents p i p_{i} , and the approach is based on variational methods, including the use of the Nehari manifold. Additionally, global existence results for the parabolic counterpart of the system are explored, with an emphasis on the potential well methods.
In this paper we deal with the bounded critical points of a Riesz energy of attractive-repulsive type in dimension 1. Under suitable assumptions on the growth of the kernel in the origin, we are able to prove that they are continuous inside their support.
We consider a variational approach to solve parabolic problems by minimising a functional over time and space. To achieve existence results, we investigate the notion of 𝒜 {\mathscr{A}} -quasiconvexity for non-homogeneous operators in anisotropic spaces. The abstract theory is then applied to formulate a variational solution concept for the generalised Navier–Stokes equations.
This paper deals with the dynamics - driven by the gradient flow of negative fractional seminorms - of empirical measures towards equi-spaced ground states. Specifically, we consider periodic empirical measures mu on the real line that are screened by the Lebesgue measure, i.e., with mu - dx having zero average. To each of these measures mu we associate a (periodic) function u satisfying u ' = dx - mu. For s is an element of(0, 1/2) we introduce energy functionals E-s(mu) that can be understood as the density of the s-Gagliardo seminorm of u per unit length. Since for s >= 1/2, the s-Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For s is an element of[1/2, 1) we define E-epsilon(s)(mu) := E-s(mu(epsilon)), where mu(epsilon) is obtained by mollifying mu on scale epsilon. We prove that the minimizers of E-s and E-epsilon(s) are the equi-spaced configurations of particles with lattice spacing equal to one. Then we prove the exponential convergence of the corresponding gradient flows to the equi-spaced steady states. Finally, although for s is an element of[1/2, 1) the energy functionals E-epsilon(s) blow up as epsilon -> 0, their gradients are uniformly bounded (with respect to epsilon), so that the corresponding trajectories converge, as epsilon -> 0, to the gradient flow solution of a suitable renormalized energy.
We study a vectorial L infinity {L<^>{\infty}} -variational problem of second order, where the supremal functional depends on the vector function u through a linear elliptic operator in divergence form. We prove existence and uniqueness of the minimiser u infinity {u_{\infty}} under prescribed Dirichlet boundary conditions, together with a characterisation of u infinity {u_{\infty}} as solution of a specific system of PDEs. Our result can be seen as a twofold extension of the one in [N. Katzourakis and R. Moser, Existence, uniqueness and structure of second order absolute minimisers, Arch. Ration. Mech. Anal. 231 2019, 3, 1615-1634]: We generalise it to the vectorial setting and, at the same time, we consider more general elliptic operators in place of the Laplacian.
We study free-discontinuity functionals in nonlinear elasticity, where discontinuities correspond to the phenomenon of cavitation. The energy comprises two terms: a volume term accounting for the elastic energy; and a surface term concentrated on the boundaries of the cavities in the deformed configuration that depends on their unit normal. First, we prove the existence of energy-minimizing deformations. While the treatment of the volume term is standard, that of the surface term relies on the regularity of inverse deformations, their weak continuity properties, and Ambrosio's lower semicontinuity theorem for special functions of bounded variation. Additionally, we identify sufficient conditions for minimality by employing outer variations and applying the formula for the first derivative of the anisotropic perimeter.
This paper concerns a class of elliptic systems of p-Laplace type with complex-valued coefficients and source terms. We extend the real-valued theory of the elliptic p-Laplace equation to the complex-valued case. We establish the existence and uniqueness of solutions to the Dirichlet problem and prove the Schauder estimate in the case of H & ouml;lder continuous coefficients and source terms. We also consider families of coefficient functions parametrized by a complex variable and prove a differentiability result for the map taking the complex parameter to the corresponding solution.
We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard prescribed-mean-curvature functional. We prove that in a ball centred at the singularity there exists a sequence of smooth CMC hypersurfaces, with the same prescribed mean curvature, that converge to the given one. Moreover, these hypersurfaces arise as boundaries of minimisers. In ambient dimension 8 the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt–Simon approximation theorem.)
We provide a counter-example to Hutchinson's original proof of C-1,C-alpha representation of curvature m-varifolds with L-q-integrable second fundamental form and q > m in [J. E. Hutchinson, C-1,C-alpha multiple function regularity and tangent cone behaviour for varifolds with second fundamental form in L-p, Geometric Measure Theory and the Calculus of Variations (Arcata 1984), Proc. Sympos. Pure Math. 44, American Mathematical Society, Providence 1986, 281-306]. We also provide an alternative proof of the same result and introduce a method of decomposing varifolds into nested components preserving weakly differentiability of a given function. Furthermore, we prove the structure theorem for curvature varifolds with null second fundamental form which is widely used in the literature.
We begin by characterizing metabelian distributions in terms of principal bundle structures. Then, we prove that in sub-Riemannian manifolds with metabelian distributions of rank r, the projection of strictly singular trajectories to some r-dimensional manifold must remain within an analytic variety. As a consequence, for rank-2 metabelian distributions, geodesics are of class C^1.
We establish the Liouville theorem for positive constant sigma(k)-curvature equation in R-+(n) and positive constant boundary B-k(g) curvature equation, where the boundary curvature B-k(g) is discovered by Sophie Chen in [S.-Y. S. Chen, Conformal deformation on manifolds with boundary, Geom. Funct. Anal. 19 (2009), no. 4, 1029-1064] from the natural variational functional for sigma(k)(A(g)).
We study the gradient regularity of solutions to measure data elliptic systems with Uhlenbeck-type structure and Orlicz growth. For any bounded Borel measure, pointwise estimates for the gradient of solutions are provided in terms of the truncated Riesz potential. This allows us to show a precise transfer of regularity from data to solutions on various scales.
We perform a dimension reduction analysis for a coupled rate-dependent/rate-independent adhesive-contact model in the setting of visco-elastodynamic plates. We work with a weak solvability notion inspired by the theory of (purely) rate-independent processes, and accordingly term the related solutions `Semistable Energetic'. For Semistable Energetic solutions, the momentum balance holds in a variational sense, whereas the flow rule for the adhesion parameter is replaced by a semi-stability condition coupled with an energy-dissipation inequality. Prior to addressing the dimension reduction analysis, we show that Semistable Energetic solutions to the three-dimensional damped adhesive contact model converge, as the viscosity term tends to zero, to three-dimensional Semistable Energetic solutions for the undamped corresponding system. We then perform a dimension reduction analysis, both in the case of a vanishing viscosity tensor, and in the complementary setting in which the damping is assumed to go to infinity as the thickness of the plate tends to zero. In both regimes, the presence of adhesive contact yields a nontrivial coupling of the in-plane and out-of-plane contributions. In the vanishing-viscosity case we additionally confine the analysis to the case in which also inertia is neglected: in the vanishing-thickness limit we thus obtain purely rate-independent evolution for the adhesive contact phenomenon, still formulated in terms of the Semistable Energetic solution concept. In the second, undamped scenario, inertia is instead encompassed, thus the limiting evolution retains a mixed rate-dependent/rate-independent character, and is again given in terms of an energy-dissipation inequality and a semistability condition.
A representation formula for the solution of the ∞ {\infty} -Laplace equation is constructed in a punctured square, the prescribed boundary values being u = 0 {u=0} on the sides and u = 1 {u=1} at the centre. This so-called ∞ {\infty} -potential is obtained with a hodograph method. The heat equation is used and one of Jacobi’s Theta functions appears. The formula disproves a conjecture.
Let alpha is an element of R. In this paper, we study Plateau's problem for the two-dimensional parametric energy integral (B) | X |(alpha) | X (u) boolean AND X (v) | du dv. Given a closed rectifiable Jordan curve Gamma subset of R-3 contained in a ball (B) over bar (R) ((m) over bar) of center (m) over bar is an element of R-3 and radius R > 0, we prove the existence of a conformally parametrized analytic minimizer spanning Gamma. The value of R and the point (m) over bar depend on alpha. We also prove necessary conditions for the existence of multiply connected stationary surfaces having prescribed disconnected boundaries.
We examine the interior regularity of solutions to a degenerate normalized p-Laplace equation, where the degeneracy is governed by a modulus of continuity whose inverse satisfies a Dini continuity condition. We prove that under very general assumptions on the degeneracy law, solutions belong to the C 1 {C<^>{1}} class. We argue by approximating the solutions by a sequence of hyperplanes, which allows us to prove the desired regularity.
