Related papers: Schauder Estimates for Germs by Scaling
The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems.
Based on the special properties of Liouville eigenoperators a perturbation theory for the partition sum is given. It is applicable for any temperature and includes the case of degenerate Hamiltonians. To demonstrate the reliability of the…
Let $\Omega$ be a flat torus and $G$ be the green's function of $-\Delta$ on $\Omega$. One intriguing mystery of $G$ is how the number of its critical points is related to blowup solutions of certain PDEs. In this article we prove that for…
We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive…
Complex solutions to squared Bessel SDEs appear naturally in relation to Schramm-Loewner evolutions. We prove a large deviation principle for such solutions as the dimension parameter tends to $-\infty$.
We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results…
Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and…
We prove a dispersive estimate for the evolution of Schroedinger operators H = -\Delta + V(x) in three dimensions. The potential should belong to the closure of bounded compactly-supported functions with respect to the golbal Kato norm.…
We give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant elliptic problems, including Lane-Emden and Schr\"odinger type systems. This applies to various classes of…
We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schr\"odinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral…
In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schr\"odinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of…
We give a necessary and sufficient condition for the maximum principle of Schr\"{o}dinger operators in terms of the bottom of the spectrum of time-changed processes. As a corollary, we obtain a sufficient condition for the Liouville…
In the last 20 years we have accumulated an enormous amount of data on elementary particles and their interactions. This data serves two purposes: to fix the phenomenological parameters of the Standard Model [SM] and to verify that the SM…
We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein-de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we…
In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic…
Using results of Gathmann, we prove the following theorem: If a smooth projective variety X has generically semisimple (p,p)-quantum cohomology, then the same is true for the blow-up of X at any number of points. This a successful test for…
We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is…
We consider semidiscrete finite differences schemes for the periodic Scr\"odinger equation in dimension one. We analyze whether the space-time integrability properties observed by Bourgain in the continuous case are satisfied at the…
The representations of the kernels of the transmutation operator and of its inverse relating the one-dimensional Schr\"odinger operator with the second derivative are obtained in terms of the eigenfunctions of a corresponding…
We consider the nonlinear Schr\"{o}dinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same…