Related papers: Comments on the quantitative uniqueness of continu…
We prove an existence and uniqueness result for solutions to linear $X$-elliptic equations with $L^1$ data and zero Dirichlet boundary conditions. Such solutions depend continuously on the datum. Moreover, we show that an improvement in the…
Recent advances in quantitative unique continuation properties for solutions to uniformly elliptic, divergence form equations (with Lipschitz coefficients) has led to a good understanding of the vanishing order and size of singular and zero…
We study properties of positive functions satisfying (E) --$\Delta$u + u p -- M |$\nabla$u| q = 0 is a domain $\Omega$ or in R N + when p > 1 and 1 < q < min{p, 2}. We concentrate our research on the solutions of (E) vanishing on the…
We investigate slowly converging solutions for non-linear evolution equations of elliptic or parabolic type. These equations arise from the study of isolated singularities in geometric variational problems. Slowly converging solutions have…
In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong…
For a mean field game system, we prove the unique continuation which asserts that if Cauchy data are zero on arbitrarily chosen lateral subboundary, then the solution identically vanishes.
Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations…
We provide fine asymptotics of solutions of fractional elliptic equations at boundary points where the domain is locally conical; that is, corner type singularities appear. Our method relies on a suitable smoothing of the corner singularity…
Motivated by infinite-dimensional optimal control problems with endpoint state constraints, in this Note, we introduce the notion of finite codimensional exact controllability for evolution equations. It is shown that this new…
We consider stochastic non-linear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational…
Existence and uniqueness of mass-conserving classical solutions to the continuous coagulation equation with collisional breakage are investigated for an unbounded class of collision kernels and a particular case of the distribution…
We prove the existence of solutions to the Cauchy-Dirichlet problem associated with a class of fully nonlinear anisotropic evolution equations. We prove a comparison principle and conclude the uniqueness of solutions. All results are…
For general second order evolution equations, we prove an optimal condition on the degree of unboundedness of the damping, that rules out finite-time extinction. We show that control estimates give energy decay rates that explicitly depend…
The main result of the present paper consists in a quantitative estimate of unique continuation at the boundary for solutions to the wave equation. Such estimate is the sharp quantitative counterpart of the following strong unique…
First, a new sufficient condition for uniqueness of weak solutions is proved for the system of 2D viscous Primitive Equations. Second, global existence and uniqueness are established for several classes of weak solutions with partial…
In this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials. We derive lower bounds of decay rate at infinity for any nontrivial solution under some…
In this paper, we study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other…
We establish existence of probabilistically strong solutions and pathwise uniqueness for a class of quasilinear stochastic evolution equations on bounded domains. Our results combine recent weak existence results for quasilinear stochastic…
This survey on stationary and evolutionary problems with gradient constraints is based on developments of monotonicity and compactness methods applied to large classes of scalar and vectorial solutions to variational and quasi-variational…
We consider a variational problem with boundary singularity and Dirichlet condition. We give a blow-up analysis for sequences of solutions of an equation with exponential nonlinearity. Also, we derive a compactness criterion under some…