Related papers: On sharp Agmon-Miranda maximum principles
Given a subgroup $\Gamma$ of rational points on an elliptic curve $E$ defined over ${\mathbf Q}$ of rank $r \ge 1$ and any sufficiently large $x \ge 2$, assuming that the rank of $\Gamma$ is less than $r$, we give upper and lower bounds on…
We start in this paper a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and…
In this paper, we prove the necessary and sufficient maximum principles (NSMPs in short) for the optimal control of systems described by a quasilinear stochastic heat equation within convex control domains, which all the coefficients…
We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on…
By using the unique continuation principle for linear elliptic systems, we can simplify the proof of a recent variational maximum principle due to Alikakos and Fusco. At the same time, this approach allows us to relax an assumption from the…
In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of "k" eigenvalues of the Hessian. In particular we shed some light on some very…
The aim of this note is twofold. Firstly, we prove an abstract version of the Calder\'on transference principle for inequalities of admissible type in the general commutative multilinear and multiparameter setting. Such an operation does…
In this work, we propose and study a new approach to formulate the optimal control problem of second-order differential equations, with a particular interest in those derived from force-controlled Lagrangian systems. The formulation results…
In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and…
An important yet challenging problem in numerical linear algebra is finding a principal submatrix with maximum determinant from a given symmetric positive semidefinite matrix. This problem arises in experimental design, statistics, and…
This paper deals with an improvement of the "a-priori stability bounds" on the variation of the action variables and on the stability time obtained from a given Birkhoff normal form around the elliptic equilibrium point of an Hamiltonian…
We investigate the regularity of the viscosity solutions to a class of degenerate/singular fully nonlinear elliptic equations with Hamiltonian terms. To overcome the difficulty caused by the simultaneous presence of the general…
In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent…
Whether there is similarity between two physical processes in the movement of objects and the complexity of behavior is an essential problem in science. How to seek similarity through the adoption of quantitative and qualitative research…
For a smooth $k$-dimensional submanifold $\Sigma$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(\Sigma)$ restriction bounds of Burq-G\'erard-Tzvetkov -- originally proved for individual Laplace--Beltrami…
In this paper, we investigate the monotonicity of solutions for a nonlinear equations involving the fractional Laplacian with variable exponent. We first prove different maximum principles involving this operator. Then we employ the direct…
In the paper we discuss two questions about smooth expanding dynamical systems on the circle. (i) We characterize the sequences of asymptotic length ratios which occur for systems with H\"older continuous derivative. The sequences of…
We establish a spatial gradient maximum principle for classical solutions to the initial and Neumann boundary value problem of some quasilinear parabolic equations on smooth convex domains.
We provide a version of the transference principle. It says that certain optimization problems for functions on the circle, the interval, and the line have the same answers. In particular, we show that the sharp constants in the…
We study the validity of the comparison and maximum principles, and their relation with principal eigenvalues, for a class of degenerate nonlinear operators that are extremal among operators with one dimensional fractional diffusion.