While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In this article, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems.
In this short note, we prove the existence of solutions to a Monge-Ampère equation of entire type derived by a weighted version of the classical Minkowski problem.
We study the Cauchy problem of the spatially homogenous fractional Kramers-Fokker-Planck equation and show that the solution enjoys Gevrey regularity and decay estimation with an L2 initial datum for positive time.
Local and global well-posedness of the coagulation-fragmentation equation with size diffusion are investigated. Owing to the semilinear structure of the equation, a semigroup approach is used, building upon generation results previously derived for the linear fragmentation-diffusion operator in suitable weighted $L^1$-spaces.
In this article, we prove global existence of classical solutions to the incompressible isotropic Hookean elastodynamics in three-dimensional thin domain $Ω_δ=\mathbb{R}^2\times [0,δ]$ with periodic boundary condition.
We establish well-posedness conclusions for the Cauchy problem associated to the dispersion generalized Zakharov-Kutnesov equation in bi-periodic Sobolev spaces $H^{s}\left(\mathbb{T}^{2}\right)$, $s>(\frac{3}{2}-\frac{1}{2^{α+2}})(\frac{3}{2}-\fracβ{4})$.
Based on the recently proved Khavinson conjecture, we establish an inequality of Schwarz-Pick type for harmonic functions on the unit ball of $\mathbb{R}^n$.
We study a Dynamic Programming Principle related to the $p$-Laplacian for $1 < p < \infty$. The main results are existence, uniqueness and continuity of solutions.
In this paper, we give the boundeness of solutions to Fractional Laplacian Ginzburg-Landau equation, which extends the Brezis theorem into the nonlinear Fractional Laplacian equation. A related linear fractional Schrodinger equation is also studied.
We prove the local existence for the Water Waves equations with large bathymetric variations on a time interval of size 1/ε, where $ε$ measures the amplitude of the wave. We just need the presence of surface tension.
We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.
We obtain the existence, regularity, uniqueness of the non-stationary problems of a class of non-Newtonian fluid is a power law fluid with $p>9/5$ in the half-space under slip boundary conditions.
In this note we present a way to approximate the Steiner problem by a family of elliptic energies of Modica-Mortola type, with an additional term relying on the weighted geodesic distance which takes care of the connexity constraint.
By using a perturbation technique in critical point theory, we prove the existence of solutions for two types of nonlinear equations involving fractional differential operators.
We consider the equation - Δu + b \nabla u = 0. The dependence of the local regularity of a solution on the properties of the coefficient b is investigated.
We consider the Korteweg-de Vries Equation (KdV) on the real line, and prove that the smooth solutions satisfy a-priori local in time $H^s$ bound in terms of the $H^s$ size of the initial data for $s\geq -4/5$.
In this paper we derive various sufficient conditions on the pressure for vanishing velocity in the incompressible Navier-Stokes and the Euler equations in $\Bbb R^N$.