On Variational Models for Quasi-static Bingham Fluids

We study a quasi-static incompressible flow of Bingham type with constituent law \[ \begin{array}{ll} T = p\left| {\cal E}u\right| ⁁{p-2}{\cal E}u+\beta \frac{{\cal E}u}{\left| {\cal E}u\right| } & \text{if }{\cal E}u\neq 0, \\ \left| T\right| \leq \beta & \text{if }{\cal E}u = 0, \end{array} \] T = p∣ℰu∣^p-2ℰu+β ℰu ∣ℰu∣ if ℰu≠0, ∣T∣⩽β if ℰu = 0, where p≥2 and β>0. Here ℰu denotes the strain velocity and T the corresponding stress. The problem admits a variational formulation in the sense that the velocity field u minimizes the energy I(u) = ∫_Ω∣ℰu∣^p+β∣ℰu∣dx in the space {v∈H^1,p(Ω,ℝⁿ): div v = 0} subject to appropriate boundary conditions. We then show smoothness of u on the set {x∈Ω: ℰu≠0}.     

On a Free Boundary Problem

This paper considers a discontinuous semilinear elliptic problem: \[ -\Delta u=g(u)H(u-\mu )\quad \text{in }\Omega,\qquad u=h\quad \text{on }% \partial \Omega, \] −Δu=g(u)H(u−μ) in Ω, u=h on ∂Ω, where H is the Heaviside function, μ a real parameter and Ω the unit ball in ℝ². We deal with the existence of solutions under suitable conditions on g, h, and μ. It is shown that the free boundary, i.e. the set where u=μ, is sufficiently smooth.     

Uniform decay estimates for solutions of the Yamabe equation

We study positive solutions u of the Yamabe equation c_m Du-s( x) u+k( x) u^\fracm+2m-2=0{c_{m} \Delta u-s\left( x\right) u+k\left( x\right) u^{\frac{m+2}{m-2}}=0}, when k(x) > 0, on manifolds supporting a Sobolev inequality. In particular we get uniform decay estimates at infinity for u which depend on the behaviour at infinity of k, s and the L ^Γ-norm of u, for some G 3 \tfrac2mm-2{\Gamma\geq\tfrac{2m}{m-2}}. The required integral control, in turn, is implied by further geometric conditions. Finally we give an application to conformal immersions into the sphere.     

Remark on the anisotropic prescribed mean curvature equation on arbitrary domains

In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on ${x \in \varOmega \subset \mathbb {R}^n}In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on x ? \varOmega ì \mathbb Rⁿ{x \in \varOmega \subset \mathbb {R}^n}, the height u of the hypersurface M = graph u over \varOmega{\varOmega} and the unit normal N to M at (x, u). We give a condition relating H and the mean curvature of ?\varOmega{\partial \varOmega} that guarantees the existence of smooth solutions even for not necessarily convex domains.     

A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II

In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197–220, 2008) where we developed a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory. There we proved local monotonicity formulae for solutions u^e{{u^\varepsilon}} to the singular perturbation problem and for u=limu^e{u=\lim{u^\varepsilon}} , assuming that both u^e{{u^\varepsilon}} and u were defined in an arbitrary domain D{\mathcal{D}} in \mathbbR^N+1{\mathbb{R}^{N+1}} . In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while u^e{{u^\varepsilon}} are not. We derive such global formulae from a local one that we prove here. In particular, we obtain a global monotonicity formula for blow up limits u ₀ of limit functions u that are not globally defined. As a consequence of this formula, we characterize blow up limits u ₀ in terms of the value of a density at the blow up point. We also present applications of the results in this paper to the study of the regularity of ∂{u > 0} (the flame front in combustion models). The fact that our results hold for the inhomogeneous singular perturbation problem allows a very wide applicability, for instance to problems with nonlocal diffusion and/or transport.     

Optimal estimates for blowup rate and behavior for nonlinear heat equations

We first describe all positive bounded solutions of where \input amstex \loadmsbm $(y,s)\in \Bbb R^N\times \Bbb R$ , 1 < p, and (N − 2)p ≤ N + 2. We then obtain for blowup solutions u(t) of uniform estimates at the blowup time and uniform space-time comparison with solutions of u′ = u^p. © 1998 John Wiley & Sons, Inc.     

On the Existence of Multiple Periodic Solutions for the Vector <Emphasis Type="Italic">p</Emphasis>-Laplacian via Critical Point Theory

We study the vector p-Laplacian

Existence and regularity properties of non-isotropic singular elliptic equations

We establish existence and sharp regularity results for solutions to singular elliptic equations of the order u ^−β , 0 < β < 1, with gradient dependence and involving a forcing term λ f(x, u). Our approach is based on a singularly perturbed technique. We show that if the forcing parameter λ > 0 is large enough, our solution is positive. For λ small solutions vanish on a nontrivial set and therefore they exhibit free boundaries. We also establish regularity results for the free boundary and study the asymptotic behavior of the problem as b\searrow 0{\beta\searrow 0} and b\nearrow 1{\beta\nearrow 1}. In the former, we show that our solutions u _β converge to a C ^1,1 function which is a solution to an obstacle type problem. When b\nearrow 1{\beta\nearrow 1} we recover the Alt-Caffarelli theory.     

A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.     

Critical window for the vacant set left by random walk on random regular graphs

We consider the simple random walk on a random d ‐regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time \begin{align*}\left\lfloor un \right\rfloor\end{align*}, where u > 0 is a fixed positive parameter. It was shown in ?erný et al., (Ann Inst Henri Poincaré Probab Stat 47 (2011) 929–968) that this so‐called vacant set exhibits a phase transition at u = u_?: there is a giant component if u < u_? and only small components when u > u_?. In this paper we show the existence of a critical window of size n^‐1/3 around u_?. In this window the size of the largest cluster is of order n^2/3. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation

On global transformations of ordinary differential equations of the second order

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form

Refined asymptotic expansions for nonlocal diffusion equations

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form u _t = J * u – u in the whole with an initial condition u(x, 0) = u ₀(x). Under suitable hypotheses on J (involving its Fourier transform) and u ₀, it is proved an expansion of the form

A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}

We study the Cauchy problem in \mathbbR^N{\mathbb{R}^N} for the parabolic equation

Isolated singularities of nonlinear parabolic inequalities

We study C ^2,1 nonnegative solutions u(x,t) of the nonlinear parabolic inequalities

Remarks on the total variation of the Jacobian

The total variation TV(u) of the Jacobian determinant of nonsmooth vector fields u has recently been studied in [2] [3]. We focus on the subclass u(x) = φ(x/|x|) of homogeneous extensions of smooth functions In the case n = 2, we explicitely compute TV(u) for some relevant examples exhibiting a gap with respect to the total variation |Det Du| of the distributional determinant. We then provide examples of functions with |DetDu| = 0 and TV(u) = + ∞. We finally show that this gap phenomenon doesn’t occur if n ≥ 3.     

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C^∞, and let g_ij = δ_ij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W^2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C³ away from a certain singular set Σ and C^∞ away from a larger singular set Σ ∪ Σ₀. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ₀. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc.     

On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations

We construct an a priori estimate of the seminorm á u_xx ñ_{a, [`(W)]} {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} for solutions to the problem

Bounds and monotonicity for the generalized Robin problem

We consider the principal eigenvalue λ ₁^Ω(α) corresponding to Δu = λ (α) u in W, \frac?u?v = au \Omega, \frac{\partial u}{\partial v} = \alpha u on ∂Ω, with α a fixed real, and W ì Rⁿ\Omega \subset {\mathcal{R}}^n a C ^0,1 bounded domain. If α > 0 and small, we derive bounds for λ ₁^Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.