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Let (M, g) be a compact riemannian manifold of dimension >. We consider two Paneitz–Branson type equations with general coefficients and where A g and B g are smooth symmetric (2, 0)-tensors, >, > and ε is a small positive parameter. Under suitable assumptions, we construct solutions > to (E.1) and (E.2) which blow up at one point of the manifold w...
We give a sharp upper bound on the vanishing order of solutions to the Schrodinger equation with C-1 electric and magnetic potentials on a compact smooth manifold. Our main result is that the vanishing order of nontrivial solutions to Delta u + V . del u + Wu = 0 is everywhere less than C (1 + parallel to W parallel to(1/2)(C1) + parallel to V para...
Let (M, g) be a compact riemannian manifold of dimension n >= 5. We consider a Paneitz-Branson type equation with general coefficients Delta(2)(g)u - div(g)(A(g)du) + hu = vertical bar u vertical bar(2)*(-2-epsilon)u on M, where A(g) is a smooth symmetric (2, 0)-tensor, h is an element of C-infinity(M), 2* = 2n/n-4 and epsilon is a small positive p...
On a closed manifold, we give a quantitative Carleman estimate on theSchrödinger operator. We then deduce quantitative uniqueness results for solutions to the Schrödinger equation using doubling estimates.Finally we investigate the sharpness of this results with respect to the electric potential.
We study the possible vanishing order of solutions to Schrodinger equation in case the potential is a bounded function. We give an upper bound on compact manifold. We also show that this result is sharp.
We give an upper bound for the $(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic Riemannian manifolds. This is the analog of H. Donnely and C. Fefferman result on nodal set of eigenfunctions.