Zhiren Jin

Liouville theorems for harmonic maps

SummaryWe prove several Liouville theorems for harmonic maps between certain classes of Riemannian manifolds. In particular, the results can be applied to harmonic maps from the Euclidean space (Rm,g0) to a large class of Riemannian manifolds. Our assumptions on the harmonic maps concern the asymptotic behavior of the maps at ∞.

Principal Eigenvalues with Indefinite Weight Functions

Both existence and non-existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved. The existence of a continuous family of principal eigenvalues is demonstrated.

Multiple solutions for a class of semilinear elliptic equations

We show that for a class of semilinear elliptic equations there are at least three nontrivial solutions. Existence of infinitely many solutions is also shown when the nonlinear term is odd. In our results, the nonlinear term can grow super-critically at infinity.

Theconvergence Rate of Solutions of Quasilinear Elliptic Equations in Slabs

Solutions of Dirichlet problems for quasilinear elliptic equations in unbounded domains inside a slab are considered. The rate at which solutions converge to their limiting functions at infinity is established in terms of properties of the top order coefficients of the operator and the rate at which the boundary values converge to their limiting functions.Our proofs are based on constructing appropriate barrier functions which depend on the behavior of coefficients of the operator and the rate of convergence of boundary value

A MAXIMUM PRINCIPLE FOR SOLUTIONS OF A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS ON UNBOUNDED DOMAINS

ABSTRACT For solutions on unbounded domains of boundary value problems for a class of quasilinear elliptic equations which are not uniformly elliptic, we prove that the solutions have the same bounds as those of the boundary data.

Convergent Rates for Solutions of Dirichlet Problems of Quasilinear Equations

The convergent rates for bounded solutions of Dirichlet problems of quasilinear elliptic (possibly degenerate) equations in slab-like domains are derived in terms of the convergent rates of the boundary data and the coefficients of the operator. The equations considered include the prescribed mean curvature equation. The results are proved by constructing a family of local barrier functions based on the structure of the operator and the convergent rate of the boundary data. The construction of local barriers is inspired by early work due to Finn and Serrin that is related to the minimal surface equation.