Uniqueness and comparison principles for semilinear equations and inequalities in Carnot groups

Variants of the Kato inequality are proved for distributional solutions of semilinear equations and inequalities on Carnot groups. Various applications to uniqueness, comparison of solutions and Liouville theorems are presented.

Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

In this paper, we give the boundedness of solutions to Ginzburg–Landau fractional Laplacian equation, which extends the Herve–Herve theorem into the nonlinear fractional Laplacian equation. We follow Brezis’ idea to use the Kato inequality. A related linear fractional Schrödinger equation is also studied.

The Hijazi Inequalities on Complete RiemannianSpincManifolds

We extend the Hijazi type inequality, involving the energy-momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spincmanifolds without boundary and of finite volume. Under some additional assumptions, using the refined Kato inequality, we prove the Hijazi type inequality for elements of the essential spectrum. The limiting cases are also studied.

Stability of the two-dimensional Brown–Ravenhall operator

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.

Stability of the two-dimensional Brown–Ravenhall operator

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.