1,720,981 research outputs found
Singular metrics and associated conformal groups underlying differential operators of mixed and degenerate types
No full textFor partial differential equations of mixed elliptic-hyperbolic and degenerate types which are the Euler-Lagrange equations for an
associated Lagrangian, we examine an associated metric structure which becomes singular on the hypersurface where the operator degenerates. In particular, we show that the ``non-trivial part'' of the complete symmetry group for the differential operator
(calculated in a previous paper by D. Lupo and K. R. Payne [Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types. Duke Math. J. (2005)]) corresponds to a group of local conformal transformations with respect to the metric away from the metric singularity and that the group extends smoothly across the singular surface. In this way, we define and calculate the conformal group for these operators as well as for lower order singular perturbations which are defined naturally by
the singular metric
PDE of Mixed Type: The Twin Challenges of Globalization and Diversity
Full text linkPartial Differential Equations (PDE) of mixed elliptic-hyperbolic type arise in particular but interesting
contexts such as transonic fluid flow and isometric embeddings of Riemannian manifolds whose curvature changes sign.
Problems which involve mixed type PDE are difficult due in large measure to diversity; that is, the mixture of qualitative types
competes with the fact that sharp PDE tools are often calibrated to the type of the equation.
The most interesting and natural problems involve, of course, nonlinear equations, but progress on them remains inhibited
due to a glaring lack of precise information on linear equations of mixed type. For example, even for linear equations, the
question of what constitutes a well posed boundary value problem is particularly
delicate as the question of desired regularity is crucial. Moreover, spectral theory is almost completely absent which complicates the
treatment of natural problems which possess a variational structure with associated functionals which are strongly
indefinite. For truly nonlinear problems, handling possible singularities or shocks is a main objective.
As is typical for PDE problems, one often can reduce the question at hand to the presence of suitable a priori estimates.
For mixed type equations, such estimates, even when locally available, need not be globalizable in a robust or clear-cut way.
We will give a general overview of some of the interesting problems which involve mixed type PDE as well as some strategies for obtaining global information
Multiplier methods for mixed type equations
No full textWe present a survey of recent results on
existence, uniqueness and non-existence for boundary value problems for equations of mixed elliptic-hyperbolic type. The common technical feature is the use of suitable integral identities and estimates that arise from well chosen multipliers which are the infinitesimal generator of an invariance or almost invariance of the differential equation
Weak well posedness of the Dirichlet problem for equations of mixed ellptic-hyperbolic type
Full text linkEquations of mixed elliptic-hyperbolic type with a homogeneous Dirichlet condition imposed on the entire boundary will be discussed. Such closed problems are typically overdetermined in spaces of classical solutions in contrast to the well-posedness for classical solutions that can result from opening the boundary by prescribing the boundary condition only on a proper subset of the boundary. Closed problems arise, for example, in models of transonic fluid flow about a given profile, but very little is known on the well-posedness in spaces of weak solutions. We present recent progress, obtained in collaboration with D. Lupo and C.S. Morawetz, on the well-posedness in weighted Sobolev spaces as well as the beginnings of a regularity theory.<br /
On the dirichlet problem of mixed type for lower hybrid waves in axisymmetric cold plasmas
Full text linkFor a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on the weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms
A dual variational approach to a class of nonlocal semilinear Tricomi problems.
No full textThe existence of at least one nontrivial solution to a class of semi- linear Tricomi problems is established via an application of the dual variational method which captures the solution as the preimage of a minimum of a suitable dual action functional. The boundary conditions are homogeneous Dirichlet conditions on a suitable part of the boundary, as dictated by uniqueness theorems for the linear problem. While there are good compactness properties for the inverse operator for the linear problem, there is a manifest asymmetry in the linear part due to the form of the boundary conditions. The linear part is symmetrized by introducing suitable reflection operators on symmetric domains, which then results in a nonlocal character of the nonlinearity
Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types
No full textFor 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
Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction
No full textAbstractFor second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L2-based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue
On locally essentially bounded divergence measure fields and sets of locally finite perimeter
Full text linkChen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss-Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol'pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss-Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way
The sonic line as a free boundary
No full textWe 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
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