We prove a \(Γ\)-convergence result for a diffeomorphism-natural discrete MDL-type functional to the Einstein-Hilbert action with the Gibbons-Hawking-York boundary term. On boundary-fitted, shape-regular meshes we establish interior and boundary blow-ups, identify the Carathéodory densities \(f_{\mathrm{in}}=α_0+α_1 R\) and \(f_{\mathrm{bdry}}=β_1 K\), and obtain the \(\liminf/\limsup\) bounds via a recovery sequence based on reflected Fermi smoothing. A boundary first-layer asymptotics shows that boundary cells contribute at order \(h^{d-1}\), yielding a global \(O(h)\) boundary remainder, while the interior remainder is \(O(h^2)\). The paper is foundational; Appendix~E specifies a reproducible protocol for rate checks and calibration of \(α_0,α_1,β_1\).
We study localization, pinning, and mobility in the fractional discrete nonlinear Schrödinger equation (fDNLS) with generalized power-law coupling. A finite-dimensional spatial-dynamics reduction of the nonlocal recurrence yields onsite and offsite stationary profiles; their asymptotic validity, orbital stability of onsite solutions, and $\ell^2$ proximity to the exact lattice solutions are established. Using the explicit construction of localized states, it is shown that the spatial tail behavior is algebraic for all $α$ > 0. The Peierls-Nabarro barrier (PNB) is computed, and the parameter regimes are identified where it nearly vanishes; complementary numerical simulations explore mobility/pinning across parameters and exhibit scenarios consistent with near-vanishing PNB. We also analyze modulational instability of plane waves, locate instability thresholds, and discuss the role of nonlocality in initiating localization. Finally, we establish small-data scattering, and quantify how fDNLS dynamics approximates the nearest-neighbor DNLS on bounded times while exhibiting distinct global behavior for any large $α$.
A dual variational principle is defined for the nonlinear system of PDE describing the dynamics of dislocations in elastic solids. The dual variational principle accounting for a specified set of initial and boundary conditions for a general class of PDE is also developed.
We investigate the large-distance asymptotics of optimal Hardy weights on $\mathbb Z^d$, $d\geq 3$, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar $\frac{(d-2)^2}{4}|x|^{-2}$ as $|x|\to\infty$. We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on $\mathbb Z^d$: (1) averages over large sectors have inverse-square scaling, (2), for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3), for i.i.d.\ coefficients, there is a matching inverse-square lower bound on moments. The results imply $|x|^{-4}$-scaling for Rellich weights on $\mathbb Z^d$. Analogous results are also new in the continuum setting. The proofs leverage Green's function estimates rooted in homogenization theory.
We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.
Following the classical result of long-time asymptotic convergence towards the Gaussian kernel that holds true for integrable solutions of the Heat Equation posed in the Euclidean Space $\mathbb{R}^n$, we examine the question of long-time behaviour of the Heat Equation in the Hyperbolic Space $\mathbb{H}^n$, $n>1$, also for integrable solutions. We show that the typical convergence proof towards the fundamental solution works in the class of radially symmetric solutions. We also prove the more precise result that says that this limit behaviour is exactly described by the 1D Euclidean kernel, but only after correction of a remarkable outward drift with constant speed produced by the geometry. Finally, we find that such fine convergence results are false for general nonnegative solutions with integrable initial data.
We study the interior Bernoulli free boundary problem for the infinity Laplacian. Our results cover existence, uniqueness, and characterization of solutions (above a threshold representing the "infinity Bernoulli constant"), their regularity, and their relationship with the solutions to the interior Bernoulli problem for the $p$-laplacian.
This is a companion paper to our recent work [9], where we studied the interior Bernoulli free boundary for the infinity Laplacian. Here we consider its variational side, which corresponds to the supremal version of the Alt--Caffarelli minimization problem.
We study the tail asymptotic of sub-exponential probability densities on the real line. Namely, we show that the n-fold convolution of a sub-exponential probability density on the real line is asymptotically equivalent to this density times n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular sub-exponential functions and use it to find an analogue of Kesten's bound for functions on $\mathbb{R}^d$. The results are applied for the study of the fundamental solution to a nonlocal heat-equation.
In Dacorogna-Moser theorem on the pullback equation $\varphi^* (g)=f$ between two prescribed volume forms (with the same total volume), control of support of the solutions can be obtained from that of the initial data, while keeping optimal regularity. This result answers a problem implicitly raised on page 14 of Dacorogna-Moser's original article ("On a partial differential equation involving the Jacobian determinant", Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 1-26), and fully generalizes the solution to the particular case of $g\equiv 1$ (prescribed Jacobian PDE, $\text{det}\,\nabla\varphi=f$) given in the author's paper "Dacorogna-Moser theorem on the Jacobian determinant equation with control of support", Discrete Cont. Dyn. Syst. 37 (2017), 4071-4089.
We describe the macroscopic behavior of evolutions by crystalline curvature of planar sets in a chessboard--like medium, modeled by a periodic forcing term. We show that the underlying microstructure may produce both pinning and confinement effects on the geometric motion.
We consider a weakly coupled system of discounted Hamilton--Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific solution of the limit system as the discount factor goes to zero. The analysis is based on a generalization of the theory of Mather minimizing measures for Hamilton--Jacobi systems and on suitable random representation formulae for the discounted solutions.
In this paper we prove a nonautonomous chain rule formula for the distributional divergence of the composite function $\boldsymbol{v}(x)=\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a divergence--measure vector field and $u$ is a function of bounded variation. As an application, we prove a uniqueness result for scalar conservation laws with discontinuous flux.
We prove that the unique solution to the Dirichlet problem with constant source term for the inhomogeneous normalized infinity Laplacian on a convex domain of $\mathbb{R}^N$ is of class $C^1$. The result is obtained by showing as an intermediate step the power-concavity (of exponent $1/2$) of the solution.
Graziano Crasta, Virginia De Cicco, Guido De Philippis
We prove a uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.
We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any $ε_0>0$, an $Ø(λ^{-ε_0})$ quasimode must have $L^2$ mass in the "wings" bounded below by $λ^{-2-δ}$ for any $δ>0$. The proof uses the author's recent work on 0-Gevrey smooth domains to approximate quasimodes on $C^{1,1}$ domains. There is an improvement for $C^{2,α}$ domains.
We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict subsolutions of class $\CC^{1,1}$ in $\R^N$. The proofs are based on the use of Lax--Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set.
We prove a uniqueness result of solutions for a system of PDEs of Monge-Kantorovich type arising in problems of mass transfer theory. The results are obtained under very mild regularity assumptions both on the reference set $Ω\subset\mathbf{R}^n$, and on the (possibly asymmetric) norm defined in $Ω$. In the special case when $Ω$ is endowed with the Euclidean metric, our results provide a complete description of the stationary solutions to the tray table problem in granular matter theory.
We adapt the metric approach to the study of stationary ergodic Hamilton-Jacobi equations, for which a notion of admissible random (sub)solution is defined. For any level of the Hamiltonian greater than or equal to a distinguished critical value, we define an intrinsic random semidistance and prove that an asymptotic norm does exist. Taking as source region a suitable class of closed random sets, we show that the Lax formula provides admissible subsolutions. This enables us to relate the degeneracies of the critical stable norm to the existence/nonexistence of exact or approximate critical admissible solutions.