Global Estimates for Quasilinear Elliptic Equations on Reifenberg Flat Domains

New global regularity estimates are obtained for solutions to a class of quasilinear elliptic boundary value problems. The coefficients are assumed to have small BMO seminorms, and the boundary of the domain is sufficiently flat in the sense of Reifenberg. The main regularity estimates obtained are in weighted Lorentz spaces. Other regularity results in Lorentz–Morrey, Morrey, and Hölder spaces are shown to follow from the main estimates.     

Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry

Journal of Dynamics and Differential Equations - We study existence and multiplicity of positive ground states for the scalar curvature equation $$begin{aligned} varDelta u+ K(|x|),...     

Quasilinear Equations for Viscoelasticity of Strain-Rate Type

We consider the quasilinear m ×  m system of partial differential equations that governs the motion of a viscoelastic material of strain-rate type on a bounded domain in . The dependence of the stress on both the strain and strain-rate is nonlinear, and our hypotheses allow for a potential energy which is a nonconvex function of the strain. The critical hypothesis is that the dependence of the stress function on the strain rate is uniformly strictly monotone (in the sense of Minty and Browder). We prove the existence and uniqueness of weak solutions to a natural initial-boundary value problem for a large class of constitutive functions. We then treat the question of H ²-regularity of solutions and show that, while regularity in the initial data is preserved, solutions do not, in general, become more regular than their initial data. This generalizes a result for the semilinear case due to Rybka.     

Nonclassical Potential Symmetry Generators of Differential Equations

We determine the nonclassical potential symmetries for a number ofequations that arise in the literature. A large number of these areobtained for some equations which only admit a single potential(classical) symmetry (e.g., the wave equation and the motion of wavesthrough some medium). However, we show that some of the exact solutionsinvariant under the nonclassical potential symmetries are equivalent toknown solutions but these solutions are not obtainable through theclassical point or potential symmetries. The Korteweg–deVries equation,it is shown, does not admit nonclassical potential symmetries – as inthe classical case.     

Radial Symmetry for Elliptic Boundary-Value Problems on Exterior Domains

By the Alexandroff-Serrin method [2, 14] of moving hyperplanes we obtain radial symmetry for the domain and the solutions of on an exterior domain , subject to the overdetermined boundary conditions , on , at and in . In particular, the following conjecture from potential theory due to P. Gruber (cf. [11, 8]) is proved: Let or be a bounded smooth domain with a constant source distribution on and let be the induced single-layer potential. If is constant in , then is a ball. (Accepted January 29, 1996)     

The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations

The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems.     

Radon Measure-Valued Solutions for a Class of Quasilinear Parabolic Equations

Initial value problems for quasilinear parabolic equations having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. In contrast, it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the Radon measures of the initial data. We call such solutions measure-valued, in contrast to function-valued solutionspreviously considered in the literature. We first show that there is a natural notion of measure-valued solution of problem (P) below, in spite of its nonlinear character. A major consequence of our definition is that, if the space dimension is greater than one, the concentrated part of the solution with respect to the Newtonian capacity is constant in time. Subsequently, we prove that there exists exactly one solution of the problem, such that the diffuse part with respect to the Newtonian capacity of the singular part of the solution (with respect to the Lebesgue measure) is concentrated for almost every positive time on the set where “the regular part (with respect to the Lebesgue measure) is large”. Moreover, using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time. Finally, the regularity problem is addressed, as we give conditions (depending on the space dimension, the initial data and the rate of convergence at infinity of the nonlinearity ψ) to ensure that the measure-valued solution of problem (P) is, in fact, function-valued.     

Geometric Approach to Nonvariational Singular Elliptic Equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms, as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter (γ - 1), for 0 < γ < 1, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases.We establish existence as well as sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and we obtain fine geometric-measure properties of the free boundary ${\mathfrak{F} = \partial{u > 0}}$ . In particular, we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region {u > 0} and the ${\mathcal{H}^{n-1}}$ almost-everywhere weak differentiability property of ${\mathfrak{F}}$ .     

Degenerate Conformally Invariant Fully Nonlinear Elliptic Equations

We establish Liouville theorems for , entire solutions and locally Lipschitz entire weak solutions to general degenerate conformally invariant fully nonlinear elliptic equations of second order. For applications to local gradient estimates of solutions of general conformally invariant fully nonlinear elliptic equations of second order, see [20].     

Removable Singularities for Fully Nonlinear Elliptic Equations

We obtain a removability result for the fully nonlinear uniformly elliptic equations F(D ² u)+f(u)=0. The main theorem states that every solution to the equation in a punctured ball (without any restrictions on the behaviour near the centre of the ball) is extendable to the solution in the entire ball provided the function f satisfies certain sharp conditions depending on F. Previously such results were known for linear and quasilinear operators F. In comparison with the semi- or quasilinear theory the techniques for the fully nonlinear equations are new and based on the use of the viscosity notion of generalised solution rather than the distributional or the weak solutions. Accepted May 3, 2000?Published online November 16, 2000     

A Concentration Phenomenon for Semilinear Elliptic Equations

For a domain ${\Omega \subset \mathbb{R}^{N}}$ we consider the equation $$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$ with zero Dirichlet boundary conditions and ${p\in(2, 2^*)}$ . Here ${V \geqq 0}$ and Q _n are bounded functions that are positive in a region contained in ${\Omega}$ and negative outside, and such that the sets {Q _n  > 0} shrink to a point ${x_0 \in \Omega}$ as ${n \to \infty}$ . We show that if u _n is a nontrivial solution corresponding to Q _n , then the sequence (u _n ) concentrates at x ₀ with respect to the H ¹ and certain L ^q -norms. We also show that if the sets {Q _n  > 0} shrink to two points and u _n are ground state solutions, then they concentrate at one of these points.     

Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

Contact Symmetry Algebras of Scalar Ordinary Differential Equations

Using Lie's classification of irreducible contact transformations in thecomplex plane, we show thata third-order scalar ordinary differential equation (ODE)admits an irreducible contact symmetry algebra if and only if it is transformableto q ⁽³⁾=0 via a local contact transformation. This result coupled with the classification of third-order ODEs with respect to point symmetriesprovide an explanation of symmetry breaking for third-order ODEs. Indeed, ingeneral, the point symmetry algebra of a third-order ODE is not asubalgebra of the seven-dimensional point symmetry algebra of q ⁽³⁾=0.However, the contact symmetry algebra of any third-order ODE, except forthird-order linear ODEs with four- and five-dimensional pointsymmetry algebras, is shown to be a subalgebra of the ten-dimensional contact symmetryalgebra of q ⁽³⁾==0. We also show that a fourth-orderscalar ODE cannot admit an irreducible contact symmetry algebra. Furthermore, weclassify completely scalar nth-order (n5) ODEs which admitnontrivial contact symmetry algebras.     

Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint

We prove the existence of solutions for a quasi-variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. We also obtain the existence of stationary solutions by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results.     

Lipschitz Regularity for Elliptic Equations with Random Coefficients

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale L ^∞-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (for example finite range of dependence). We also prove a quenched L ² estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

Quantitative Stochastic Homogenization of Elliptic Equations in Nondivergence Form

We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge–Ampère equation.