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Strong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schrödinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.

A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove the Liouville theorem in some particular cases. Finally, we give an alternative form of the stress–energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.

In this paper, we study the properties of $k$-plurisubharmonic functions defined on domains in ${ \mathbb{C} }^{n} $. By the monotonicity formula, we give an alternative proof of the weak continuity of complex $k$-Hessian operators with respect to local uniform convergence.

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On the fractional powers of a Schrödinger operator with a Hardy-type potential

Gradient estimates for semilinear elliptic systems and other related results

WEAK CONTINUITY OF THE COMPLEX $k$-HESSIAN OPERATORS WITH RESPECT TO LOCAL UNIFORM CONVERGENCE

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3 results

On the fractional powers of a Schrödinger operator with a Hardy-type potential

Gradient estimates for semilinear elliptic systems and other related results

WEAK CONTINUITY OF THE COMPLEX $k$-HESSIAN OPERATORS WITH RESPECT TO LOCAL UNIFORM CONVERGENCE