Searcharxiv⌕ Search

arXiv subjects

Scott N. Armstrong

Publications and source records attributed to Scott N. Armstrong.

32 records · Page 2Linked to original sources
arXiv2011

Singular solutions of fully nonlinear elliptic equations and applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén-Lindelöf result as well as a principle of positive singularities in certain Lipschitz domains.

math.AP↗
arXiv2011

Partial regularity of solutions of fully nonlinear uniformly elliptic equations

We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is $C^{2,α}$ on the compliment of a closed set of Hausdorff dimension at most $ε$ less than the dimension. The equation is assumed to be $C^1$, and the constant $ε> 0$ depends only on the dimension and the ellipticity constants. The argument combines the $W^{2,ε}$ estimates of Lin with a result of Savin on the $C^{2,α}$ regularity of viscosity solutions which are close to quadratic polynomials.

math.AP↗
arXiv2011

A new approach to Liouville theorems for elliptic inequalities

In this article we review a new method for proving the nonexistence of positive solutions of elliptic inequalities in unbounded domains in $\rn$, which was recently introduced by the authors. We expose our method and new results on the two most frequently encountered cases: inequalities involving the p-Laplacian or the Pucci extremal operator.

math.AP↗
arXiv2011

Unique continuation for fully nonlinear elliptic equations

We show that a viscosity solution of a uniformly elliptic, fully nonlinear equation which vanishes on an open set must be identically zero, provided that the equation is $C^{1,1}$. We do not assume that the nonlinearity is convex or concave, and thus \emph{a priori} $C^2$ estimates are unavailable. Nevertheless, we use the boundary Harnack inequality and a regularity result for solutions with small oscillations to prove that the solution must be smooth at an appropriate point on the boundary of the set on which it is assumed to vanish. This then permits us to conclude with an application of a classical unique continuation result for linear equations.

math.AP↗
arXiv2010

Nonexistence of positive supersolutions of elliptic equations via the maximum principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of $\mathbb{R}^n$. The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the $p$-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.

math.AP↗
arXiv2010

Convexity criteria and uniqueness of absolutely minimizing functions

We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary values under minimal assumptions on $H.$ Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in $L^\infty.$

math.AP↗
arXiv2010

Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities

We study fully nonlinear elliptic equations such as \[ F(D^2u) = u^p, \quad p>1, \] in $\R^n$ or in exterior domains, where $F$ is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of $F$, that sharply characterizes the range of $p>1$ for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-Véron \cite{B} as well as Cutri and Leoni \cite{CL}, who found critical exponents for supersolutions in the whole space $\R^n$, in case $-F$ is Laplace's operator and Pucci's operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle.

math.AP↗
arXiv2009

An infinity Laplace equation with gradient term and mixed boundary conditions

We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation \[ -Δ_\infty u - β|Du| = f, \] subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation.

math.AP↗
arXiv2009

Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions $Φ$ and $\tilde Φ$ of the PDE \[ F(D^2Φ) = 0 \quad {in} \mathbb{R}^n \setminus \{0 \} \] for any positively homogeneous, uniformly elliptic operator $F$. Corresponding to $F$ are two unique scaling exponents $α^*, \tildeα^* > -1$ which describe the homogeneity of $Φ$ and $\tilde Φ$. We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation $F(D^2u) = 0$, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of $F(D^2u) = 0$ in $\mathbb{R}^n \setminus \{0 \}$ which are bounded on one side in a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two-player differential game.

math.AP↗
arXiv2009

Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation \[ u_t + F(D^2u) = 0 \quad {in} \R^n\times \R_+, \] for a positively homogeneous operator $F$, subject to the initial condition $u(x,0) = g(x)$, under the assumption that $g$ does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution $Φ^+$ and negative solution $Φ^-$, which satisfy the self-similarity relations \[ Φ^\pm (x,t) = λ^{α^\pm} Φ^\pm (λ^{1/2} x, λt). \] We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to $Φ^+$ ($Φ^-$) locally uniformly in $\R^n \times \R_+$. The anomalous exponents $α^+$ and $α^-$ are identified as the principal half-eigenvalues of a certain elliptic operator associated to $F$ in $\R^n$.

math.AP↗
arXiv2009

The Dirichlet problem for the Bellman equation at resonance

We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and positively homogeneous. Examples of such operators include the Hamilon-Jacobi-Bellman operator and the Pucci extremal operators. In the case that the two principal half-eigenvalues are not equal, we show that the measures which achieve the minimum in this formula provide a partial characterization of the solvability of the corresponding Dirichlet problem at resonance.

math.AP↗
arXiv2008

Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a smooth bounded domain $Ω$, under the assumption the nonlinearity $F$ is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal "half" eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the "second" eigenvalue is positive, and generalize the anti-maximum principle of Clément and Peletier to homogeneous, fully nonlinear operators.

math.AP↗
arXiv2007

Solvability of Symmetric Word Equations in Positive Definite Letters

Let $S(X,B)$ be a symmetric (``palindromic'') word in two letters $X$ and $B$. A theorem due to Hillar and Johnson states that for each pair of positive definite matrices $B$ and $P$, there is a positive definite solution $X$ to the word equation $S(X,B)=P$. They also conjectured that these solutions are finite and unique. In this paper, we resolve a modified version of this conjecture by showing that the Brouwer degree of such an equation is equal to 1 (in the case of real matrices). It follows that, generically, the number of solutions is odd (and thus finite) in the real case. Our approach allows us to address the more subtle question of uniqueness by exhibiting equations with multiple real solutions, as well as providing a second proof of the result of Hillar and Johnson in the real case.

math.OA↗