Archive for Rational Mechanics and Analysis最新文献

We provide an algebraic framework to describe renormalization in regularity structures based on multi-indices for a large class of semi-linear stochastic PDEs. This framework is “top-down”, in the sense that we postulate the form of the counterterm and use the renormalized equation to build a canonical smooth model for it. The core of the construction is a generalization of the Hopf algebra of derivations in Linares et al. (Commun Am Math Soc 3:1–64, 2023, https://doi.org/10.1090/cams/16), which is extended beyond the structure group to describe the model equation via an exponential map; this allow us to implement a renormalization procedure which resembles the preparation map approach in our context.

We study stochastic homogenisation problems for free discontinuity functionals under a new assumption on the surface terms, motivated by cohesive fracture models. The results are obtained using a characterization of the limit functional by means of the asymptotic behaviour of suitable minimisation problems on cubes with very simple boundary conditions. An important role is played by the subadditive ergodic theorem.

We study certain “geometric-invariant resonant cavities” introduced by Liberal, Mahmoud, and Engheta in a 2016 Nature Communications paper. They are cylindrical devices modeled using the transverse magnetic reduction of Maxwell’s equations, so the mathematics is two-dimensional. The cross-section consists of a dielectric inclusion surrounded by an “epsilon-near-zero” (ENZ) shell. When the shell has just the right area, its interaction with the inclusion produces a resonance. Mathematically, the resonance is a nontrivial solution of a 2D divergence-form Helmoltz equation (nabla cdot left( varepsilon ^{-1}(x,omega ) nabla u right) + omega ^2 mu u = 0), where (varepsilon (x,omega )) is the (complex-valued) dielectric permittivity, (omega ) is the frequency, (mu ) is the magnetic permeability, and a homogeneous Neumann condition is imposed at the outer boundary of the shell. This is a nonlinear eigenvalue problem, since (varepsilon ) depends on (omega ). Use of an ENZ material in the shell means that (varepsilon (x,omega )) is nearly zero there, so the PDE is rather singular. Working with a Lorentz model for the dispersion of the ENZ material, we put the discussion of Liberal et. al. on a sound foundation by proving the existence of the anticipated resonance when the loss parameter of the Lorentz model is sufficiently small. Our analysis is perturbative in character, using the implicit function theorem despite the apparently singular form of the PDE. While the existence of the resonance depends only on the area of the ENZ shell, its quality (that is, the rate at which the resonance decays) depends on the shape of the shell. It is therefore natural to consider an associated optimal design problem: what shape shell gives the slowest-decaying resonance? We prove that if the dielectric inclusion is a ball then the optimal shell is a concentric annulus. For an inclusion of any shape, we study a convex relaxation of the design problem using tools from convex duality. Finally, we discuss the conjecture that our relaxed problem amounts to considering homogenization-like limits of nearly optimal designs.

We provide a rigidity statement for the equality case of the Heintze–Karcher inequality in substatic manifolds. We apply such a result in the warped product setting to fully remove assumption (H4) in the celebrated Brendle’s characterization of constant mean curvature hypersurfaces in warped products.

We study Lipschitz critical points of the energy (int _Omega g(det text {D} u) ,text {d} x) in two dimensions, where g is a strictly convex function. We prove that the Jacobian of any Lipschitz critical point is constant, and that the Jacobians of sequences of approximately critical points converge strongly. The latter result answers, in particular, an open problem posed by Kirchheim, Müller and Šverák in 2003.

In this paper, we prove the global existence of analytic solution for 3D anisotropic Navier-Stokes system with initial data which is small and analytic in the vertical variable. We shall also prove that this solution will be analytic in the horizontal variables soon after (t>0.) Furthermore, we show that the ratio between the analytic radius, (R_textrm{h}(t),) of the solution in the horizontal variables and ( sqrt{t}) satisfies (lim _{trightarrow 0_+}frac{R_textrm{h}(t)}{sqrt{t}}=infty .)

Image inpainting involves filling in damaged or missing regions of an image by utilizing information from the surrounding areas. In this paper, we investigate a fully nonlinear partial differential equation inspired by the modified Cahn–Hilliard equation. Instead of using standard potentials that depend solely on pixel intensities, we consider morphological image enhancement filters that are based on a variant of the shock filter:

This is referred to as the Shock Filter Cahn–Hilliard Equation. The equation is nonlinear with respect to the highest-order derivative, which poses significant mathematical challenges. To address these, we make use of a specific approximation argument, establishing the existence of a family of approximate solutions through the Leray–Schauder fixed point theorem and the Aubin–Lions lemma. In the limit, we obtain a solution strategy wherein we can prove the existence and uniqueness of solutions. Proving the latter involves the Kruzhkov entropy type-admissibility conditions. Additionally, we use a numerical method based on the convexity splitting idea to approximate solutions of the nonlinear partial differential equation and achieve fast inpainting results. To demonstrate the effectiveness of our approach, we apply our method to standard binary images and compare it with variations of the Cahn–Hilliard equation commonly used in the field.

We consider one-dimensional self-similar solutions to the isentropic Euler system when the initial data are at vacuum to the left of the origin. For (x>0), the initial velocity and sound speed are of the form (u_0(x)=u_+x^{1-lambda }) and (c_0(x)=c_+x^{1-lambda }), for constants (u_+in mathbb {R}), (c_+>0), (lambda in mathbb {R}). We analyze the resulting solutions in terms of the similarity parameter (lambda ), the adiabatic exponent (gamma ), and the initial (signed) Mach number (text {Ma}=u_+/c_+). Restricting attention to locally bounded data, we find that when the sound speed initially decays to zero in a Hölder manner ((0<lambda <1)), the resulting flow is always defined globally. Furthermore, there are three regimes depending on (text {Ma}): for sufficiently large positive (text {Ma})-values, the solution is continuous and the initial Hölder decay is immediately replaced by (C^1)-decay to vacuum along a stationary vacuum interface; for moderate values of (text {Ma}), the solution is again continuous and with an accelerating vacuum interface along which (c^2) decays linearly to zero (i.e., a “physical singularity”); for sufficiently large negative (text {Ma})-values, the solution contains a shock wave emanating from the initial vacuum interface and propagating into the fluid, together with a physical singularity along an accelerating vacuum interface. In contrast, when the sound speed initially decays to zero in a (C^1) manner ((lambda <0)), a global flow exists only for sufficiently large positive values of (text {Ma}). The non-existence of global solutions for smaller (text {Ma})-values is due to rapid growth of the data at infinity and is unrelated to the presence of a vacuum.

We give a complete and rigorous derivation of the mechanical energy for twisted 2D bilayer heterostructures without any approximation beyond the existence of an empirical many-body site energy. Our results apply to both the continuous and discontinuous continuum limit. Approximating the intralayer Cauchy–Born energy by linear elasticity theory and assuming an interlayer coupling via pair potentials, our model reduces to a modified Allen–Cahn functional. We rigorously control the error, and, in the case of sufficiently smooth lattice displacements, provide a rate of convergence for twist angles satisfying a Diophantine condition.

In this paper, we establish a (C^{1,alpha })-regularity theorem for almost-minimizers of the functional (mathcal {F}_{varepsilon ,gamma }=P-gamma P_{varepsilon }), where (gamma in (0,1)) and (P_{varepsilon }) is a nonlocal energy converging to the perimeter as (varepsilon ) vanishes. Our theorem provides a criterion for (C^{1,alpha })-regularity at a point of the boundary which is uniform as the parameter (varepsilon ) goes to 0. Since the two terms in the energy are of the same order when (varepsilon ) is small, we are considering here much stronger nonlocal interactions than those considered in most related works. As a consequence of our regularity result, we obtain that, for (varepsilon ) small enough, volume-constrained minimizers of (mathcal {F}_{varepsilon ,gamma }) are balls. For small (varepsilon ), this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel G with sufficiently fast decay at infinity.