Form boundedness of the general second‐order differential Operator

By Vladimir G. Maz'ya, Igor Verbitsky. 2004

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We give explicit necessary and sufficient conditions for the boundedness of the general second‐order differential operator $${\cal L} = \sum\limits_{i,\,j=1}^{n} a_{ij} \partial_{i} \partial_{j} + \sum\limits_{j=1}^{n} b_{j} \partial_{j} + c$$ with real‐ or complex‐valued distributional coefficients aij, bj, and c, acting from the Sobolev space W1, 2(ℝn) to its dual W−1, 2(ℝn). This enables us to obtain analytic criteria for the fundamental notions of relative form boundedness, compactness, and infinitesimal form boundedness of ℒ︁ with respect to the Laplacian on L2(ℝn).

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d997fb5cb5689a6e45b03be9323ee5e6faa5332e

Citations in the corpus

• Sharp convex Lorentz–Sobolev inequalities (2011, unknown): “For more connections of (3) to Sobolev-type embeddings, see [26,55,57–60,62,76].”