# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

January 2015 , Volume 14 , Issue 1

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2015, 14(1): i-i doi: 10.3934/cpaa.2015.14.1i +[Abstract](1010) +[PDF](91.2KB)
Abstract:
This special issue Emerging Trends in Nonlinear PDE of this journal was conceived during the Fall 2013 research semester “Evolutionary Problems” held at the Mittag-Leffler-Institute and its colophon conference Quasilinear PDEs and Game Theory held at Uppsala University in early December. The editors of this special issue participated in these activities. Following several conversations with other participants, we solicited manuscripts from participants of the Mittag-Leffler special semester, the Uppsala conference, as well as from colleagues working in closely related fields. Seventeen papers (out of twenty-one) are authors by participants in the research program at the Mittag-Leffler or the conference at Uppsala.
2015, 14(1): 1-21 doi: 10.3934/cpaa.2015.14.1 +[Abstract](1724) +[PDF](486.2KB)
Abstract:
In this paper, we study the potential theoretic aspects of the normalized $p$-Laplacian evolution, see (1.1) below. A systematic study of such equation was recently started in [1], [4] and [25]. Via the classical Perron approach, we address the question of solvability of the Cauchy-Dirichlet problem with "very weak" assumptions on the boundary of the domain. The regular boundary points for the Dirichlet problem are characterized in terms of barriers. For $p \geq 2$, in the case of space - time cylinder $G \times (0,T)$, we show that $(x,t) \in \partial G \times (0, T]$ is a regular boundary point if and only if $x \in \partial G$ is a a regular boundary point for the p-Laplacian. This latter operator is the steady state corresponding to the evolution (1.1) below. Consequently, when $p\geq 2$ the Cauchy- Dirichlet problem for (1.1) can be solved in cylinders whose section is regular for the $p$-Laplacian. This can be thought of as an analogue of the results obtained in [17] for the standard parabolic $p$-Laplacian div$(|Du|^{p-2}Du) - u_t = 0$.
2015, 14(1): 23-49 doi: 10.3934/cpaa.2015.14.23 +[Abstract](1774) +[PDF](523.0KB)
Abstract:
We consider non-homogeneous, singular ($0 < m < 1$) porous medium type equations with a non-negative Radon-measure $\mu$ having finite total mass $\mu(E_T)$ on the right-hand side. We deal with a Cauchy-Dirichlet problem for these type of equations, with homogeneous boundary conditions on the parabolic boundary of the domain $E_T$, and we establish the existence of a solution in the sense of distributions. Finally, we show that the constructed solution satisfies linear pointwise estimates via linear Riesz potentials.
2015, 14(1): 51-62 doi: 10.3934/cpaa.2015.14.51 +[Abstract](1582) +[PDF](390.6KB)
Abstract:
The minimizers of convex integral functionals of the form \begin{eqnarray} \mathfrak{F} (v, \Omega) = \int_{\Omega} F (Dv (x)) dx, \end{eqnarray} defined on Sobolev mappings $v$ in $W^{1,1}_{g}(\Omega R^N)$ are characterized as the energy solutions to the Euler--Lagrange system for $\mathfrak{F}$. We assume that the integrands $F: R^{N\times n} \to R$ are $C^1$, convex and super--linear at infinity, and the boundary datum $g \in W^{1,1}(\Omega, R^N)$ must satisfy $F(sDg) \in L^1(\Omega )$ for some number $s>1$.
2015, 14(1): 63-82 doi: 10.3934/cpaa.2015.14.63 +[Abstract](1387) +[PDF](472.4KB)
Abstract:
For a given bounded Lipschitz set $\Omega$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(\Omega)\hookrightarrow L^2(\partial \Omega)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are close'' to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.
2015, 14(1): 83-106 doi: 10.3934/cpaa.2015.14.83 +[Abstract](1442) +[PDF](464.9KB)
Abstract:
In this paper we introduce a method to define fractional operators using mean value operators. In particular we discuss a geometric approach in order to construct fractional operators. As a byproduct we define fractional linear operators in Carnot groups, moreover we adapt our technique to define some nonlinear fractional operators associated with the $p-$Laplace operators in Carnot groups.
2015, 14(1): 107-119 doi: 10.3934/cpaa.2015.14.107 +[Abstract](1330) +[PDF](376.4KB)
Abstract:
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast di usion equation with a critical exponent. We improve a previous result on slow convergence to Barenblatt pro les.
2015, 14(1): 121-125 doi: 10.3934/cpaa.2015.14.121 +[Abstract](1628) +[PDF](296.7KB)
Abstract:
For a very strong diffusion equation like total variation flow it is often observed that the solution stops at a steady state in a finite time. This phenomenon is called a finite time stopping or a finite time extinction if the steady state is zero. Such a phenomenon is also observed in one-harmonic map flow from an interval to a unit circle when initial data is piecewise constant. However, if the target manifold is a unit two-dimensional sphere, the finite time stopping may not occur. An explicit example is given in this paper.
2015, 14(1): 127-132 doi: 10.3934/cpaa.2015.14.127 +[Abstract](1414) +[PDF](337.8KB)
Abstract:
We investigate a version of the Phragmén--Lindelöf theorem for solutions of the equation $\Delta_\infty u=0$ in unbounded convex domains. The method of proof is to consider this infinity harmonic equation as the limit of the $p$-harmonic equation when $p$ tends to $\infty$.
2015, 14(1): 133-142 doi: 10.3934/cpaa.2015.14.133 +[Abstract](1737) +[PDF](360.9KB)
Abstract:
A comparison principle for viscosity solutions of second-order quasilinear elliptic partial di erential equations with no zeroth order terms is shown. A di erent transformation from that of Barles and Busca in [3] is adapted to enable us to deal with slightly more general equations.
2015, 14(1): 143-166 doi: 10.3934/cpaa.2015.14.143 +[Abstract](1109) +[PDF](487.9KB)
Abstract:
We prove convergence of critical points to the nonlinear elastic energies $J^h$ of 3d thin incompressible plates, to critical points of the 2d energy obtained as the $\Gamma$-limit of $J^h$ in the von Kármán scaling regime. The presence of incompressibility constraint requires to restrict the class of admissible test functions to bounded divergence-free variations on the 3d deformations. This poses new technical obstacles, which we resolve by means of introducing 3d extensions and truncations of the 2d limiting deformations, specific to the problem at hand.
2015, 14(1): 167-184 doi: 10.3934/cpaa.2015.14.167 +[Abstract](1577) +[PDF](405.8KB)
Abstract:
We provide an alternative approach to the existence of solutions to dynamic programming equations arising in the discrete game-theoretic interpretations for various nonlinear partial differential equations including the infinity Laplacian, mean curvature flow and Hamilton-Jacobi type. Our general result is similar to Perron's method but adapted to the discrete situation.
2015, 14(1): 185-199 doi: 10.3934/cpaa.2015.14.185 +[Abstract](1521) +[PDF](427.4KB)
Abstract:
In the first part of the paper we review some mean value properties and their connections to the Laplacian and other significant nonlinear operators like the $p$-Laplacian and the infinity-Laplacian. The second part is devoted to the unique continuation property, including a brief description of the methods, some of the main problems in the area and connections to the so called infinity mean value property.
2015, 14(1): 201-216 doi: 10.3934/cpaa.2015.14.201 +[Abstract](1616) +[PDF](414.6KB)
Abstract:
We prove that solutions to Cauchy problems related to the $p$-parabolic equations are stable with respect to the nonlinearity exponent $p$. More specifically, solutions with a fixed initial trace converge in an $L^q$-space to a solution of the limit problem as $p>2$ varies.
2015, 14(1): 217-228 doi: 10.3934/cpaa.2015.14.217 +[Abstract](1380) +[PDF](361.5KB)
Abstract:
We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.
2015, 14(1): 229-244 doi: 10.3934/cpaa.2015.14.229 +[Abstract](1210) +[PDF](465.1KB)
Abstract:
In this paper we deal with an optimal matching problem, that is, we want to transport two commodities (modeled by two measures that encode the spacial distribution of each commodity) to a given location, where they will match, minimizing the total transport cost that in our case is given by the sum of the two different Finsler distances that the two measures are transported. We perform a method to approximate the matching measure and the pair of Kantorovich potentials associated with this problem taking limit as $p\to \infty$ in a variational system of $p-$Laplacian type.
2015, 14(1): 245-268 doi: 10.3934/cpaa.2015.14.245 +[Abstract](1437) +[PDF](514.0KB)
Abstract:
This article deals with the following quasilinear parabolic problem \begin{eqnarray} u_t-\Delta_p u=h(x)u^{q}, u\geq 0 & \text{in} \,\, \Omega\times (0,\infty),\\ u(x,t)=0 & \text{on}\,\, \partial \Omega\times (0,\infty), \\ u(x,0)=f(x), \,\, f\geq 0 & \text{in} \,\, \Omega, \end{eqnarray} where $-\Delta_p u=-div(|\nabla u|^{p-2}\nabla u)$, $p>1$, $q>0$, $h(x)>0$ and $f(x)\geq 0$ are non negative functions satisfying suitable hypotheses. We assume the domain $\Omega$ is either a bounded regular domain or the whole $\mathbb{R}^N$. The main contribution of this work is to prove that the optimal exponent in the reaction term in order to prove existence of a global positive solution is $q_0=\min\{1,(p-1)\}$. More precisely, we obtain the following conclusions
If $1 < p < 2$ and $0 < q < p-1$, there is no finite extinction time.

If $p > 2$ and $0 < q< 1$, there is no finite speed of propagation.
In both cases the result is optimal.
2015, 14(1): 269-284 doi: 10.3934/cpaa.2015.14.269 +[Abstract](1233) +[PDF](455.7KB)
Abstract:
Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \begin{eqnarray} u_{t} - ( \frac{p}{ N + p - 2 } ) \Delta^{1}_{p} u = 0 \quad \mbox{for} \quad x \in R^N \quad \mbox{and} \quad t > 0 , \\ \\ u(\cdot,0) = u_0 \in BUC(R^N). \end{eqnarray} is given by the exponential formula \begin{eqnarray} u(t) := \lim_{n \to \infty}{ ( M^{t/n}_{p} )^{n} u_{0} } \ , \end{eqnarray} where the statistical operator $M^h_p \colon BUC( R^{N} ) \to BUC( R^{N} )$ is defined by \begin{eqnarray} (M^{h}_{p} \varphi)(x) := (1-q) median_{\partial B(x,\sqrt{2h})}{ \{ \varphi \} } + q \int_{\partial B(x,\sqrt{2h})}{ \varphi ds } \end{eqnarray} when $1 \leq p \leq 2$, with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, and by \begin{eqnarray} (M^{h}_{p} \varphi )(x) := ( 1 - q ) midrange_{\partial B(x,\sqrt{2h})}{ \{ \varphi\} } + q \int_{\partial B(x,\sqrt{2h})}{ \varphi ds } \end{eqnarray} when $p \geq 2$, with $q = \frac{ N }{ N + p - 2 }$. Possible extensions to problems with Dirichlet boundary conditions are mentioned briefly.
2015, 14(1): 285-311 doi: 10.3934/cpaa.2015.14.285 +[Abstract](1774) +[PDF](550.5KB)
Abstract:
This is a survey of some recent contributions by the authors on global integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form. Minimal assumptions on the regularity of the ground domain and of the prescribed data are pursued.
2015, 14(1): 313-327 doi: 10.3934/cpaa.2015.14.313 +[Abstract](1686) +[PDF](494.8KB)
Abstract:
For a Hamiltonian $H \in C^2(R^{N \times n})$ and a map $u:\Omega \subseteq R^n \to R^N$, we consider the supremal functional \begin{eqnarray} E_\infty (u,\Omega) := \|H(Du)\|_{L^\infty(\Omega)} . \end{eqnarray} The Euler-Lagrange" PDE associated to (1) is the quasilinear system \begin{eqnarray} A_\infty u := (H_P \otimes H_P + H[H_P]^\bot H_{PP})(Du):D^2 u = 0. \end{eqnarray} (1) and (2) are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author [28]. Herein we show that the Dirichlet problem for (2) admits for all $n = N \geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ scalar uniqueness theory of Jensen [24] has no counterpart when $N \geq 2$. The key idea in the proofs is to recast (2) as a first order differential inclusion $Du(x) \in \mathcal{K} \subseteq R^{n\times n}$, $x\in \Omega$.
2015, 14(1): 329-339 doi: 10.3934/cpaa.2015.14.329 +[Abstract](1481) +[PDF](378.9KB)
Abstract:
We address the question if eigenfunctions of the 1-Laplacian, which are obtained through a variational argument, are also viscosity solutions of the associated strongly degenerate formal Euler equation. The answer is positive, but examples show also that there are many more viscosity solutions than expected.
2015, 14(1): 341-360 doi: 10.3934/cpaa.2015.14.341 +[Abstract](1600) +[PDF](590.1KB)
Abstract:
The variational problem of minimizing the functional $u \mapsto \int_\Omega |Du| + \frac{1}{p}\int_\Omega |Du|^p - \int_\Omega au$ on a domain $\Omega\subset R^2$ under zero boundary values, which among other things models the laminar flow of a Bingham fluid, shows an interesting phenomenon: its minimizer has a maximum set with positive measure (a "plateau"). In this work we show properties of the minimizer and its plateau, most notably, connectedness and a lower bound of its measure. In addition we look at the related boundary value problem where $a=0$, $\Omega$ is a convex ring, and two boundary values are given. For this problem we show various results, including quasiconcavity of the minimizer and regularity.

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