Abstract We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton–Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form $$\partial _t u+H(x,t,Du,D^2u)=0$$ ∂ t u + H ( x , t , D u , D 2 u ) = 0 in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator $$\varepsilon F(x,t,D^2u)$$ ε F ( x , t , D 2 u ) . Assuming that $$u\in C^\alpha _x$$ u ∈ C x α , $$u_0\in C^\eta $$ u 0 ∈ C η , $$H\in C^\beta _x$$ H ∈ C x β and having power growth $$\gamma $$ γ in the gradient entry, we establish a convergence rate of order $$\varepsilon ^{\min \left\{ \frac{\eta }{2},\frac{\beta +\gamma (\alpha -1)}{\beta +\gamma (\alpha -1)+2-\alpha }\right\} }$$ ε min η 2 , β + γ ( α - 1 ) β + γ ( α - 1 ) + 2 - α . Our novel approach exploits the regularizing properties of sup/inf-convolutions for viscosity solutions and the comparison principle. We also obtain explicit constants and do not assume differentiability properties neither on solutions nor on H . The same method provides new convergence rates for the vanishing viscosity approximation of the stationary counterpart of the equation and for transport equations with Hölder coefficients.

ABSTRACT This paper is devoted to the quantitative homogenization of multiscale elliptic operator , where , , and . We assume that is 1‐periodic in each and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios . In the present paper, under the assumption of real analytic coefficients, we introduce the so‐called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to . This convergence rate is optimal in the sense that cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double‐log scale‐separation condition.

The strong unique continuation property of elliptic operators with interior single-point degeneration

In this study, we examine a length preserving geodesic curvature difference flow for smooth strictly horocyclically convex simple closed curves in the hyperbolic plane H2. Given an initial curve γ1 and a target curve γ2 of the same hyperbolic length, we evolve γ1 by a normal speed given by the difference of the reciprocals of geodesic curvatures evaluated at points with the same outward unit normal, together with a time-dependent scalar term Γ(t) chosen to preserve the hyperbolic length. Using Leichtweiβ’s hyperbolic support function and Howe’s curvature formula, the flow is reformulated as a quasilinear uniformly parabolic equation on S1 with a nonlocal term Γ(t). We prove short-time existence, uniqueness, and preservation of strict horocyclic convexity. Linearizing the support function equation at the target support function yields a uniformly elliptic operator whose kernel contains the infinitesimal isometry directions. Under a spectral gap assumption on a normalized slice transverse to the isometry orbit, we prove global existence and exponential convergence for initial data sufficiently close to the target curve. In the last section, this assumption is verified explicitly when the target curve is a geodesic circle.

We address the problem given by the following partial differential equation: some semi-Linear parabolic equations with uniformly elliptic non-local operators in Half-Space. Initially, we establish a generalized weighted average inequality and a maximum principle in unbounded domains, which are crucial for the sliding method. Then, we employ sliding to demonstrate the monotonicity of bounded positive solutions. In this paper, we will remove the monotonicity assumption of the kernel function $a(x)$ by using the sliding method. The techniques employed in the process of this method have applications to other problems related to uniformly elliptic operators.

We prove a Weitzenböck identity for general pairs of constant-coefficient homogeneous first-order partial differential operators, and deduce from it sufficient algebraic conditions for coerciveness and Morrey estimates under the natural 1/2 boundary conditions. Our proof of the W 1 , 2 elliptic estimate relies on the Aronszajn–Ne c ˘ as–Smith coercive estimate. For generalized strongly pseudoconvex domains, we improve the Morrey estimate to a weighted W 1 , 2 square function estimate, using a generalized Cauchy–Pompeiu reproducing formula and the T 1 theorem for singular integrals. We use Van Schaftingen's notion of cocanceling to study the generalized Levi forms appearing.

Abstract We study the hypoellipticity and solvability properties of a class of time-periodic evolution operators, with coefficients globally defined on $$\mathbb {R}^d$$ R d and growing polynomially with respect to the space variable. To this aim, we introduce a class of time-periodic weighted Sobolev spaces, whose elements are characterised in terms of suitable Fourier expansions associated with elliptic operators.

The Lefschetz number of an endomorphism of an elliptic complex is expressed in terms of regularized traces of the operators defining the endomorphism. This result is obtained under certain conditions on the wavefront sets of the operators in question. In the particular case of geometric endomorphisms of the complex, we obtain the classical Atiyah–Bott formula. As an application, we compute the Lefschetz numbers of nonlocal elliptic operators associated with an action of a finite group on a closed smooth manifold. For the de Rham complex, this gives a formula for the Lefschetz number in the cohomology of the orbit space in terms of fixed points.

In this work, we investigate a Schrödinger operator defined on a model graph containing small loops, under the assumption that the standard nonresonance condition—typically ensuring the holomorphic dependence of the resolvent for elliptic operators on graphs with short edges—is violated. Our analysis focuses on the behavior of those components of the resolvent that correspond to finite edges and small loops. It is shown that these components retain their holomorphic dependence on a small parameter characterizing the length of the loops. In contrast to the nonresonant case, however, the part of the resolvent associated with the small loops develops an additional contribution in the leading term of its Taylor expansion, which results in a certain localization of the resolvent on these loops.

Abstract Consider a second‐order elliptic operator in the half‐plane with coefficients depending only on the second coordinate. The Poisson kernel for is used in the representation of positive ‐harmonic functions, that is, solutions of . In probabilistic terms, the Poisson kernel is the density function of the distribution of the diffusion in with generator at the hitting time of the boundary. We prove that the Poisson kernel for is bell‐shaped: its th derivative changes sign times. In particular, it is unimodal and it has two inflection points (it is concave, then convex and then concave again).

General potential theories concern the study of functions which are subharmonic with respect to a suitable constraint set $\cF$ in the space of 2-jets. While interesting in their own right, general potential theories are being widely used to study fully nonlinear PDEs determined by degenerate elliptic operators $F$ acting on the space of 2-jets. We will discuss a powerful tool, the correspondence principle, which establishes the equivalence between $\cF$–subharmonics/superharmonics $u$ and admissible subsolutions/supersolutions $u$ (in the viscosity sense) of the PDE determined by every operator $F$ which is compatible with $\cF$. The crucial degenerate ellipticity often requires the operator to be restricted to a suitable constraint set $\cG$, which determines the admissibility. Applications to comparison principles by way of the duality-monotonicity-fiberegularity method will also be discussed.

Abstract Let $$\Omega $$ Ω be a bounded, smooth domain of $$\mathbb {R}^N$$ R N , $$N\ge 2$$ N ≥ 2 . In this paper, we prove some inequalities involving the first Robin eigenvalue of the p -laplacian operator. In particular, we prove an upper bound for the first Robin eigenvalue of nonlinear elliptic operators in terms of the first Dirichlet eigenvalue.

In this work, we investigate a class of degenerate Schrödinger equations associated to degenerate elliptic operators with irregular potentials on [Formula: see text] by introducing a suitable Hörmander metric [Formula: see text] and a [Formula: see text]-weight [Formula: see text]. We establish the well-posedness for the corresponding degenerate Schrödinger and degenerate parabolic equations. When the subellipticity is available on the degenerate elliptic operator we deduce spectral properties for a class of degenerate Hamiltonians. We also investigate the [Formula: see text] mapping properties for operators with symbols in the [Formula: see text] classes in the spirit of classical Fefferman’s [Formula: see text]-bounds for the [Formula: see text] calculus. Finally, within our [Formula: see text]-classes, sharp [Formula: see text]-estimates and Schatten–von Neumann properties for Schrödinger operators for Hörmander sums of squares are also investigated.

Probabilistic Representation for a Class of Exponential Elliptic Operators and Stochastic Runge-Kutta Methods for Semilinear Stiff Equations

A bstract The species cutoff is a moduli-dependent quantity signaling the onset of quantum gravitational phenomena, whose form can be oftentimes determined from higher-derivative and higher-curvature corrections within low-energy gravitational EFTs. In this work, we point out that these Wilson coefficients are eigenfunctions of an appropriate second-order elliptic operator defined over moduli space in theories with more than four supercharges. This was already known to be the case for the leading $$ {\mathcal{R}}^4 $$ R 4 -correction to the two-derivative (bosonic) action of maximal supergravity in d ≤ 10. Here, we reconsider this fact from the Swampland point of view and show how, in d = 10, 9, 8, solving a Laplace equation imposes non-trivial restrictions on the species hull vectors. We further argue that this property is also satisfied in settings with less supersymmetry. In particular, we focus on the $$ {\mathcal{R}}^4 $$ R 4 -operator in minimal supergravity theories in d = 10, 9, and on the leading $$ {\mathcal{R}}^2 $$ R 2 -term in setups with 8 supercharges in d = 6, 5, 4. Finally, we provide a symmetry-based criterion for determining when the relevant elliptic operator should be the Laplacian. A bottom-up rationale for this constraint remains to be fully understood, and we conclude by outlining some compelling possibilities.

This paper investigates the existence of nonnegative weak solutions to a class of degenerate elliptic equations with singular nonlinearities. The problem under consideration is of the form [Formula: see text] with homogeneous Dirichlet boundary conditions, where [Formula: see text] is a bounded domain, [Formula: see text], [Formula: see text], [Formula: see text] is a nonnegative element of the dual Sobolev space [Formula: see text], and [Formula: see text] is a continuous function that may blow up at zero but remains bounded at infinity. The degeneracy of the principal part, controlled by the parameter [Formula: see text], adds significant difficulty to the analysis. Using a double approximation scheme (regularizing both the degeneracy and the singularity), truncation arguments, monotonicity methods, and the Schauder fixed point theorem, we establish the existence of a solution [Formula: see text] under appropriate conditions on the data. Our main contribution lies in the simultaneous treatment of degeneracy and singularity, extending classical results to a broader class of non-uniformly elliptic operators. The proofs rely on uniform a priori estimates, compactness arguments, and a careful passage to the limit in the approximate problems.

In this paper, we establish the equivalence of weak and viscosity solutions for a homogeneous problem involving a mixed local and nonlocal elliptic operator in a bounded domain \Omega\subset\mathbb{R}^{N} with Lipschitz boundary. We employ a comparison principle and a priori variational estimates to prove that continuous weak solutions are viscosity solutions and bounded viscosity solutions that vanish outside \Omega are weak solutions. Our results are novel and new for mixed local and nonlocal operators, even for p=2 .

This article investigates the uniform null controllability problem for a system of coupled parabolic equations with a periodic oscillating coefficient. Our approach combines spectral analysis and Carleman estimates. First, we analyze the spectral properties of an elliptic operator with an oscillating coefficient to control the low frequencies. Then the system is allowed to evolve freely to achieve the required decay. In the third step, we establish a Carleman estimate that leads to a suitable observability result. By combining the three steps, we prove the uniform null controllability of the system, which is then used to homogenize the associated coupled parabolic system.

This paper presents and analyzes robust numerical algorithms for solving inverse problems for parabolic equations, specifically focusing on the determination of an unknown time-dependent source function from an integral flux condition. The study is motivated by mathematical models based on Navier-Stokes equations, particularly those exhibiting Poiseuille-type solutions. We employ a variational approach, formulating the inverse problem as the minimization of a Tikhonov regularization cost functional. Discrete approximation schemes are rigorously derived using finite volume methods in space and both backward Euler and Crank-Nicolson schemes in time. A key contribution of this work is the strict justification of the gradient formula for the cost functional by deriving the adjoint problem directly from the fully discrete scheme, rather than discretizing the continuous adjoint problem. This methodology is extended to problems involving fractional powers of elliptic operators and two-dimensional domains. Numerical experiments are conducted to compare the efficiency of Gradient Descent and Conjugate Gradient methods. The results demonstrate that the Conjugate Gradient method significantly outperforms standard gradient descent, maintaining high accuracy and convergence rates even with the inclusion of regularization terms and complex diffusion operators.

This article is concerned with strictly elliptic, second-order differential operators on a bounded Lipschitz domain in \mathbb{R}^{d} subject to certain non-local Wentzell–Robin boundary conditions. We prove that such operators generate strongly continuous semigroups on L^{2} -spaces and on spaces of continuous functions. We also provide a characterization of positivity and (sub-)Markovianity of these semigroups. Moreover, based on spectral analysis of these operators, we discuss further properties of the semigroup such as asymptotic behavior and, in the case of a non-positive semigroup, the weaker notion of eventual positivity of the semigroup.