Strong Unique Continuation Property Research Articles

Remark on the strong unique continuation property for parabolic operators

We consider solutions u = u ( x , t ) u = u(x,t) , in a neighbourhood of ( x , t ) = ( 0 , 0 ) (x,t) =(0,0) , to a parabolic differential equation with variable coefficients depending on space and time variables. We assume that the coefficients in the principal part are Lipschitz continuous and that those in the lower order terms are bounded. We prove that, if u ( ⋅ , 0 ) u( \cdot ,0) vanishes of infinite order at x = 0 x=0 , then u ( ⋅ , 0 ) ≡ 0 u( \cdot ,0) \equiv 0 .

Strong unique continuation property for elliptic systems of normal type in two independent variables

We give a result on strong unique continuation property for a certain elliptic system of first order in the two dimensional space. Two coefficient matrices are normal and commutative with each other. We assume, further, that their components are Hölder continuous and have continuous first order derivatives except at one point. Without any regularity assumptions on the eigenvalues, we can show the strong unique continuation property for a class of such systems under certain quantitative conditions on the first order derivatives. This result gives an improvement of a work by G. N. Hile and M. H. Protter in a special case.

Sharp counterexamples in unique continuation for second order elliptic equations

We construct nontrivial solutions with compact support for the elliptic equation ∆u = V u with V ∈ Lp, p < n/2 or V ∈ L w for n ≥ 3 and with V ∈ L1 for n = 2. The same method also yields nontrivial solutions with compact support for the elliptic equation ∆u = W∇u with W ∈ Lq, p < n or W ∈ Lw for n ≥ 2. For the second order elliptic equations ∆u = V u in R (1) respectively ∆u = W∇u in R (2) we define the (weak) unique continuation property (UCP) , respectively the strong unique continuation property (SUCP) as follows: Let u be a solution to (1) which vanishes in an open set. Then u = 0. (UCP) Let u be a solution to (1) which vanishes of infinite order at some point x0 ∈ R. Then u = 0. (SUCP) ∗Research partially supported by the DFG and DAAD. †Research partially supported by the NSF grants DMS-9970297, INT-9815286.

Strong unique continuation of eigenfunctions for p‐Laplacian operator

We show the strong unique continuation property of the eigenfunctions for p‐Laplacian operator in the case p &lt; N.

Strong Unique Continuation Property for the Dirac Equation

We discuss some Carleman type inequalities which lead to a strong unique continuation property for Dirac operators with potential dominated by the Coulomb singularity.

Note on the Paper “Strong Unique Continuation Property for the Dirac Equation” by De Carli and Ōkaji

Note on the Paper “Strong Unique Continuation Property for the Dirac Equation” by De Carli and Ōkaji

Zero sets of solutions to semilinear elliptic systems of first order

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an $n$-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth $(n-2)$-dimensional submanifolds. Hence it is countably $(n-2)$-rectifiable and its Hausdorff dimension is at most $n-2$. Moreover, it has locally finite $(n-2)$-dimensional Hausdorff measure. We show by example that every real number between 0 and $n-2$ actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order.

A strong unique continuation theorem for parabolic equations

The unique continuation is best understood for second order elliptic operators. The classical paper by Carleman [7] established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients. This was remarkable since it indicates that even a non-analytic solution of an elliptic equation can behave somewhat in an “analytic” manner. The powerful technique he used, the so-called “Carleman weighted inequality” has played a central role in later developments. In 1950’s, Aronszajn [4] and Cordes [11] generalized his result to higher dimensions. In recent years this classical subject caught new attention from a great number of people. Much efforts have been made to relax the smoothness hypotheses on the coefficients (see [6, 17, 43, 1, 22]). A seminal paper by Jerison and Kenig [26] has shown the unique continuation property for operators of the form ∆ + c(x ) with c ∈ L /2 loc where N ≥ 3 is the space dimension. Further improvements have been carried out in considering other classes of coefficients (Fefferman-Phong class and Kato class) [42, 8, 12, 29], in extending the result to operators with first derivative terms and variable leading coefficients [46, 47, 51, 52], and etc. (See [25, 15, 16, 28, 5, 27] and the references therein.) For second order linear parabolic operators with time-independent coefficients, the strong unique continuation property along with detailed estimates on the Hausdorff measures of nodal sets was reduced in [32, 33] to the previously established elliptic counterparts. The reduction from time-independent parabolic

Some remarks concerning unique continuation for the Dirac operator

We show that the strong unique continuation property for the Dirac operator holds if the potential belongs toLp withp = max(d,(3d − 4)/2).

Unique continuation for schrodinger operators with singular potentials

It is shown in this paper that the Schrodinger operator with a potential satisfying ∣V(x)∣ ≤ M/∣x∣2a.e. in the unit ball B has the strong unique continuation property in H2,2(B) for for n≥2.

Unique continuation for Δ+𝑣 and the C. Fefferman-Phong class

We show that the strong unique continuation property holds for the inequality | Δ u | ≤ | υ | | u | \left | {\Delta u} \right | \leq \left | \upsilon \right |\left | u \right | , where the potential υ ( x ) \upsilon (x) satisfies the C. Fefferman-Phong condition in a certain range of p p values. We also deal with the situation of u ( x ) u(x) vanishing at infinity. These are all consequences of appropriate Carleman inequalities.

Nonlinear perturbations of linear operators having nullspace with strong unique continuation property

Nonlinear perturbations of linear operators having nullspace with strong unique continuation property

Genericity of simple eigenvalues for elliptic PDE’s

The spectrum of a selfadjoint, C' linear elliptic partial differential operator on a compact manifold contains only isolated eigenvalues, each having finite multiplicity. It is sometimes the case that these multiplicities are unbounded; this is common in problems arising in applications because of the high degree of symmetry usually present. The main theorem shows that the property of having only simple eigenvalues is generic for operators obtained by varying the zeroth order part of a given operator. The purpose of this article is to provide a proof of the following theorem. Main theorem. Let M be a compact, connected C' manifold without boundary. Let L be a sel/adjoint, C' linear elliptic differential operator on M. The set Ep e Cw(M): all eigenvalues of L + p are simple} is residual in C'(M). The term elliptic operator used in this article will always mean the operator is selfadjoint, C' and linear. The main theorem can be summarized by saying that almost all elliptic operators obtained by varying the zeroth order part of a given operator have only simple eigenvalues. For terminology, see ? 1. The main theorem was first announced in [1] for second-order operators, and the proof is the one given in [2]; it uses perturbation theory. A proof using transversality theory, applicable to operators whose eigenfunctions satisfy the strong unique continuation property, has been obtained by K. Uhlenbeck [6]. The theorem is particularly useful in obtaining genericity results about the eigenfunctions (see [1] , [2], [6]). 1. Notation; outline of proof. The manifold M and the operator L will be fixed throughout as in the theorem. For standard terminology on elliptic operators, see [3], [4]. Received by the editors June 4, 1973 and, in revised form, January 23, 1974. AMS (MOS) subject classifications (1970). Primary 35P05, 35J30; Secondary 47A55, 47B25, 58G99.

Strong uniqueness for second order elliptic operators with Gevrey coefficients

We consider here the problem of strong unique continuation property at zero, for second order elliptic operators P = P (x,D) with complex coefficients. For such operators we obtain this property by means of suitable Carleman’s estimates in Gevrey classes of appropriate index. This index depends on the spread of the cone image of the principal symbol p of P evaluate at zero, that is p(0,RN {0}). Secondly by using similar techniques we deal with the strong unique continuation property in suitable Gevrey classes for some fourth order elliptic operators with real coefficients. Mathematics Subject Classification: Primary 35B60; Secondary 35J15, 35J30.