Strong Unique Continuation Property Research Articles

Unique continuation for differential inclusions

We consider the following question arising in the theory of differential inclusions: Given an elliptic set \Gamma and a Sobolev map u whose gradient lies in the quasiconformal envelope of \Gamma and touches \Gamma on a set of positive measure, must u be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where \Gamma \subset \mathbb{R}^{2\times 2} is an elliptic, smooth, closed curve. More precisely, we prove that the distance of \mathrm{D}u to \Gamma satisfies the strong unique continuation property. As a by-product, we obtain new results for non-linear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge–Ampère equation: concerning the latter, we obtain a new proof of the W^{2,1+\varepsilon} -regularity for two-dimensional solutions.

Unique continuation problem on RCD Spaces. I

In this note we establish the weak unique continuation theorem for caloric functions on compact RCD(K, 2) spaces and show that there exists an RCD(K, 4) space on which there exist non-trivial eigenfunctions of the Laplacian and non-stationary solutions of the heat equation which vanish up to infinite order at one point . We also establish frequency estimates for eigenfunctions and caloric functions on the metric horn. In particular, this gives a strong unique continuation type result on the metric horn for harmonic functions with a high rate of decay at the horn tip, where it is known that the standard strong unique continuation property fails.

Strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equationΔX2u=Vu, where ΔX=Δx+|x|2αΔy (0<α≤1), with x∈Rm,y∈Rn, denotes the Baouendi-Grushin type subelliptic operators, and the potential V satisfies the strongly singular growth assumption |V|≤c0ρ4, whereρ=(|x|2(α+1)+(α+1)2|y|2)12(α+1) is the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms.

Strong unique continuation for variable coefficient parabolic operators with Hardy type potential

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality|div(A(x,t)∇u)−ut|≤M|x|2|u|, where the coefficient matrix A is Lipschitz continuous in x and t. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends an earlier result of one of us with Garofalo and Manna for the heat operator.

A note on unique continuation of eigenfunctions for p-Laplacian operator in a bounded domain Ω in ℝ n with a potential V in L p (Ω)

The aim of this note is to study the problem − div ( | ∇ u | p − 2 ∇u ) + V | u | p − 2 u = 0 in Ω, where Ω is a bounded domain in R n and the potential V is assumed to be not equivalent to zero and lies in L p ( Ω ) . Also, we establish the strong unique continuation property of the eigenfunctions for the p-Laplacian operator in the case where V ∈ L p ( Ω ) .

Strong Unique Continuation from the Boundary for the Spectral Fractional Laplacian

We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non-homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Passing to traces, an analogous blow-up result and its consequent strong unique continuation property is deduced for the nonlocal fractional equation.

An application of generalized Morrey spaces to unique continuation property of the quasilinear elliptic equations

&lt;abstract&gt;&lt;p&gt;In this paper, we study nonnegative weak solutions of the quasilinear elliptic equation $ \text{div}(A(x, u, \nabla u)) = B(x, u, \nabla u) $, in a bounded open set $ \Omega $, whose coefficients belong to a generalized Morrey space. We show that $ \log (u + \delta) $, for $ u $ a nonnegative solution and $ \delta $ an arbitrary positive real number, belongs to $ \text{BMO}(B) $, where $ B $ is an open ball contained in $ \Omega $. As a consequence, this equation has the strong unique continuation property. For the main proof, we use approximation by smooth functions to the weak solutions to handle the weak gradient of the composite function which involves the weak solutions and then apply Fefferman's inequality in generalized Morrey spaces, recently proved by Tumalun et al. &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;]&lt;/sup&gt;.&lt;/p&gt;&lt;/abstract&gt;

SOME FUNCTION SPACES AND THEIR APPLICATIONS TO ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space or a Stummel class. We also show that the logarithm of a non-negative weak solution to a second order elliptic partial differential equation with potential in a generalized Morrey space or a Stummel class, under some assumptions, belongs to the bounded mean oscillation class. As a consequence, this elliptic partial differential equation has the strong unique continuation property. An example of an elliptic partial differential equation with potential in a Morrey space or a Stummel class which does not satisfy the strong unique continuation is presented.

The nodal set of solutions to some nonlocal sublinear problems

We study the nodal set of stationary solutions to equations of the form (-Delta )^s u = lambda _+ (u_+)^{q-1} - lambda _- (u_-)^{q-1}quad text {in }B_1, where lambda _+,lambda _->0, q in [1,2), and u_+ and u_- are respectively the positive and negative part of u. This collection of nonlinearities includes the unstable two-phase membrane problem q=1 as well as sublinear equations for 1<q<2. We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case s=1, we prove that the admissible vanishing orders can not exceed the critical value k_q= 2s/(2- q). Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that k_q< 1, we prove a remarkable difference with the local case: solutions can only vanish with order k_q and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.

Time and norm optimal controls for the heat equation with dynamical boundary conditions

In this paper, we deal with time and norm optimal control problems for the heat equation with dynamic boundary conditions. We prove the existence of admissible controls and, combining a suitable Carleman estimate for such an equation with some analyticity results, we show a strong unique continuation property. Then, with the aid of this unique continuation, bang‐bang properties and the connection between time and norm optimal controls can be established.

Quantitative unique continuation of solutions to the bi‐Laplace equations

In this paper, we prove a three‐ball inequality for y satisfying an equation of the form in some open, connected set Ω of with and . The derivation of such estimate relies on a delicate Carleman estimate for the bi‐Laplace equation and some Caccioppoli inequalities to estimate the lower‐terms. Based on three‐ball inequality, we then derive the vanishing order of y is less than , where | · |∞ means the L∞ norm, which is a quantitative version of the strong unique continuation property for y. Furthermore, under some priori assumptions on Vj and y, we prove that the nontrivial solution y satisfies the decay property around the point at infinity. In particular, if , this decaying rate can be improved to .

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

&lt;p style='text-indent:20px;'&gt;In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [&lt;xref ref-type="bibr" rid="b10"&gt;10&lt;/xref&gt;] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (ⅰ) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ⅱ) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.&lt;/p&gt;

A unique continuation property for a class of parabolic differential inequalities in a bounded domain

This article is concerned with a strong unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $ \Omega $ prescribed with some regularity and growth conditions. Our results show that the value of the solutions can be determined uniquely by its value on an arbitrary open subset $ \omega $ in $ \Omega $ at any given positive time $ T $. We also derive the quantitative nature of this unique continuation, that is, the estimate of a $ L^2(\Omega) $ norm of the initial data, which is majorized by that of solution on the bounded open subset $ \omega $ at terminal moment $ t = T $.

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials. Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.

A Strong Unique Continuation Property for the Heat Operator with Hardy Type Potential

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality |Δu-ut|≤M|x|2|u|,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} |\\Delta u - u_t| \\le \\frac{M}{|x|^2} |u|, \\end{aligned}$$\\end{document}with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.

Strong unique continuation for the Lamé system with less regular coefficients

We study the strong unique continuation property (SUCP) for the Lame system in the plane. The main contribution of our work is to prove that the SUCP holds when Lame coefficients $$(\mu ,\lambda )\in W^{2,s}(\Omega )\times L^\infty (\Omega )$$ for some $$s>4/3$$. In other words, we establish the SUCP for the Lame system in the plane when $$\lambda $$ is bounded and $$\mu $$ belongs to certain Holder classes.

Quantitative uniqueness estimates for the generalized non-stationary Stokes system

ABSTRACTWe study the local behavior of a solution to a generalized non-stationary Stokes system with singular coefficients in with . One of our main results is a bound on the vanishing order of a non-trivial solution u satisfying the generalized non-stationary Stokes system, which is a quantitative version of the (strong) unique continuation property for u. Different from the previously known results, our unique continuation result only involves the velocity field u. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial optimal three-cylinder inequalities for u.

Quantitative unique continuation for Schrödinger operators

We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for Δ+V. That is, for any non-trivial u that solves Δu+Vu=0 in some open, connected subset of Rn, we estimate the vanishing order of solutions in terms of the Lt-norm of V. Our results apply to all t>n2 and n≥3. With these maximal order of vanishing estimates, we employ a scaling argument to produce quantitative unique continuation at infinity estimates for global solutions to Δu+Vu=0. To handle V∈Lt for every t∈(n2,∞], we prove a novel Lp−Lq Carleman estimate by interpolating a known Lp−L2 estimate with a new endpoint Carleman estimate. This new Carleman estimate may also be used to establish improved order of vanishing estimates for equations with a first order term, those of the form Δu+W⋅∇u+Vu=0.

Space like strong unique continuation for sublinear parabolic equations

In this paper, we establish space like strong unique continuation property (sucp) for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established by Ruland in [Ru] for analogous elliptic sublinear equations. Similar to that in [Ru], this is accomplished via a new $L^{2}-L^{2}$ type Carleman estimate for a class of sublinear parabolic operators.

Carleman estimates for Baouendi–Grushin operators with applications to quantitative uniqueness and strong unique continuation

ABSTRACT In this paper, we establish some new Carleman estimates for the Baouendi–Grushin operators , in Equation (1). We apply such estimates to obtain: (i) an extension of the Bourgain–Kenig quantitative unique continuation and (ii) the strong unique continuation property for some degenerate sublinear equations.