## Carleman inequalities for the Dirac and Laplace operators and unique continuation.(English)Zbl 0627.35008

The Schrödinger operator $$(L+V)$$, where L is the Laplacian and V a potential, satisfies the unique continuation property if solutions of $$(L+V)u=0$$ which vanish in a non-empty, open subset of a connected set vanish identically. The Schrödinger operator is known to have the unique continuation property provided certain so-called Carleman inequalities hold. Reference is made to the large literature on these inequalities and their applications to uniqueness questions. In this paper best possible Carleman-type inequalities are obtained for the case when L is replaced by the Dirac operator D. It is shown that $$(D+V)$$ has the unique continuation property provided $$V\in L^{\gamma}_{loc}({\mathbb{R}}^ n)$$, $$\gamma =(3n-2)/2$$, $$n\geq 3$$ which offers an improvement on known results. An additional feature of the paper is that the approach adopted to the treatment of the Dirac operator enables a simpler proof to be obtained of best possible Carleman-type inequalities for the Laplace operator.
Reviewer: G.Roach

### MSC:

 35B60 Continuation and prolongation of solutions to PDEs 35B45 A priori estimates in context of PDEs 35J10 Schrödinger operator, Schrödinger equation
Full Text:

### References:

 [1] Berthier, A.-M; Georgescu, V, Sur la propriété de prolongement unique pour l’opérateur de Dirac, C. R. acad. sci. Paris Sér. A, 291, 603-606, (1980) · Zbl 0459.35079 [2] Carleman, T, Sur un problème d’unicité pour LES systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. mat. B, 26, 1-9, (1939) · Zbl 0022.34201 [3] Fefferman, C, The uncertainty principle, Bull. amer. math. soc., 9, 2, 129-206, (1983) · Zbl 0526.35080 [4] Hörmander, L, () [5] Hörmander, L, Uniqueness theorems for second-order elliptic differential equations, Comm. partial differential equations, 8, 1, 21-64, (1983) · Zbl 0546.35023 [6] Jerison, D; Kenig, C.E, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of math., 121, (1985) · Zbl 0593.35119 [7] Nagel, A; Stein, E.M, Lectures on pseudo-differential operators: regularity theorems and applications to non-elliptic problems, () · Zbl 0415.47025 [8] Simon, B, Schrödinger semigroups, Bull. amer. math. soc., 7, 3, 447-526, (1982) · Zbl 0524.35002 [9] Sogge, C, Oscillatory integrals and spherical harmonics, () · Zbl 0636.42018 [10] Stein, E.M, Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton, N.J · Zbl 0207.13501 [11] Taylor, M.E, Pseudo differential operators, (1981), Princeton Univ. Press Princeton, N.J · Zbl 0207.45402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.