Kevin R. Payne

Nonexistence of Nontrivial Solutions for Supercritical Equations of Mixed Elliptic-Hyperbolic Type

For semilinear partial differential equations of mixed elliptic-hyperbolic type with various boundary conditions, the nonexistence of nontrivial solutions is shown for domains which are suitably star-shaped and for nonlinearities with supercritical growth in a suitable sense. The results follow from integral identities of Pohožaev type which are suitably calibrated to an invariance with respect to anisotropic dilations in the linear part of the equation. For the Dirichlet problem, in which the boundary condition is placed on the entire boundary, the technique is completely analogous to the classical elliptic case as first developed by Pohožaev [34] in the supercritical case. At critical growth, the nonexistence principle is established by combining the dilation identity with another energy identity. For “open” boundary value problems in which the boundary condition is placed on a proper subset of the boundary, sharp Hardy-Sobolev inequalities are used to control terms in the integral identity corresponding to the lack of a boundary condition as was first done in [23] for certain two dimensional problems.

Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types

For partial differential equations of mixed elliptic-hyperbolic and degenerate types which are the Euler-Lagrange equations for an associated Lagrangian, invariance with respect to changes in independent and dependent variables is investigated, as are results in the classification of continuous one-parameter symmetry groups. For the variational and divergence symmetries, conservation laws are derived via the method of multipliers. The conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems, and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems.

ON THE MAXIMUM PRINCIPLE FOR GENERALIZED SOLUTIONS TO THE TRICOMI PROBLEM

A maximum/minimum principle for weighted W1,2 solutions to the Tricomi problem with L2 right hand side and homogeneous boundary data is established for normal Tricomi domains. In addition, the existence and uniqueness of such generalized solutions is established for arbitrary L2 right hand sides in normal domains which satisfy a convexity condition near the parabolic boundary points.

The sonic line as a free boundary

We consider the steady transonic small disturbance equations on a domain and with data that lead to a solution that depends on a single variable. After writing down the solution, we show that it can also be found by using a hodograph transformation followed by a partial Fourier transform. This motivates considering perturbed problems that can be solved with the same technique. We identify a class of such problems.

Spectral Theory for Linear Operators of Mixed Type and Applications to Nonlinear Dirichlet Problems

For a class of linear partial differential operators L of mixed elliptic-hyperbolic type in divergence form with homogeneous Dirichlet data on the entire boundary of suitable planar domains, we exploit the recent weak well-posedness result of [8] and minimax methods to establish a complete spectral theory in the context of weighted Lebesgue and Sobolev spaces. The results represent the first robust spectral theory for mixed type equations. In particular, we find a basis for a weighted version of the space comprised of weak eigenfunctions which are orthogonal with respect to a natural bilinear form associated to L. The associated eigenvalues {λ k } k∈ℕ are all non-zero, have finite multiplicity and yield a doubly infinite sequence tending to ± ∞. The solvability and spectral theory are then combined with topological methods of nonlinear analysis to establish the first results on existence, existence with uniqueness and bifurcation from (λ k , 0) for associated semilinear Dirichlet problems.

Multiplicity of nontrivial solutions for an asymptotically linear nonlocal Tricomi problem

where T ≡ −y@x − @y is the Tricomi operator on R2; R is the re9ection operator induced by composition with the map : : R → R2 de;ned by :(x; y)=(−x; y); f(u) is an asymptotically linear term such that f(0) = 0, and 7 is a symmetric admissible Tricomi domain (cf. De;nition 2:1 of Lupo and Payne [7]). That is, 7 is a bounded region in R2 that is symmetric with respect to the y-axis and has a piecewise C2 boundary @7=AC ∪BC ∪ , where is a symmetric with respect to the y-axis C2 arc in the elliptic region y? 0, with endpoints on the x-axis at A=(−x0; 0) and B=(x0; 0) and AC and BC are two characteristic arcs for the Tricomi operator in the hyperbolic region of negative and positive slope issuing from A and B, respectively, which meet at the point C on the y-axis. It should be noted that u ≡ 0 is always a solution to (NST). We refer the reader to [7] for the rationale concerning this problem, the interest

The dual variational method in nonlocal semilinear Tricomi problems.

We describe the results obtained in [15] and [16] concerning the use of the dual variational approach in order to prove the existence and multiplicity of solutions for a nonlocal variational Tricomi problem.

Nonexistence of nontrivial generalized solutions for 2-D and 3-D BVPs with nonlinear mixed type equations

A brief survey of known results, open problems and new contributions to the understanding of the nonexistence of nontrivial solutions to nonlinear boundary value problems (BVPs) whose linear part is of mixed elliptic-hyperbolic type is given. Crucial issues discussed include: the role of so-called critical growth of the nonlinear terms in the equation (often related to threshold values of continuous and compact embedding for Sobolev spaces in Lebesgue spaces), the role that hyperbolicity in the principal part plays in over-determining solutions with classical regularity if data is prescribed everywhere on the boundary, the relative lack of regularity that solutions to such problems possess and the subsequent importance to address nonexistence of generalized solutions.