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Medientyp:
E-Artikel
Titel:
Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations
Beteiligte:
Borisov, Denis;
Tautenhahn, Martin;
Veselić, Ivan
Erschienen:
AIP Publishing, 2017
Erschienen in:
Journal of Mathematical Physics, 58 (2017) 12
Sprache:
Englisch
DOI:
10.1063/1.5017938
ISSN:
0022-2488;
1089-7658
Entstehung:
Anmerkungen:
Beschreibung:
We consider elliptic differential operators on either the entire Euclidean space Rd or subsets consisting of a cube ΛL of integer length L. For eigenfunctions of the operator, and more general solutions of elliptic differential equations, we derive several quantitative unique continuation results. The first result is of local nature and estimates the vanishing order of a solution. The second is a sampling result and compares the L2-norm of a solution over a union of equidistributed δ-balls in space with the L2-norm on the entire space. In the case where the space Rd is replaced by a finite cube ΛL, we derive similar estimates. Particular features of our bound are that they are uniform as long as the coefficients of the operator are chosen from an appropriate ensemble, they are quantitative and explicit with respect to the radius δ, and they are L-independent and stable under small shifts of the δ-balls. Our proof applies to second order terms which have slowly varying coefficients on the relevant length scale. The results can also be interpreted as special cases of uncertainty relations, observability estimates, or spectral inequalities.