Maximum principles for boundary-degenerate second-order linear elliptic differential operators

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol vanishing locus. We obtain uniqueness and a priori maximum principle estimates for smooth solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators for partial Dirichlet or Neumann boundary conditions along the complement of the boundary vanishing locus. We also prove weak maximum principles and uniqueness for solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.Comment: 62 pages, 2 figures. Accepted for publication in Communications in Partial Differential Equations. Incorporates final galley proof corrections corresponding to published versio

Resolution of singularities and geometric proofs of the Lojasiewicz inequalities

The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of the Lojasiewicz inequalities in the special case where the function is $C^1$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary real or complex analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Lojasiewicz inequalities when a function is $C^N$ and generalized Morse-Bott of order $N \geq 3$; we gave an elementary proof of the Lojasiewicz inequalities when a function is $C^2$ and Morse-Bott in arXiv:1708.09775v4 (finite-dimensional case) and arXiv:1706.09349 (infinite-dimensional case).Comment: 28 pages, incorporating final galley proof correction

Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds

In this sequel to [arXiv:1412.4114], we prove an $L^{d/2}$ energy gap result for Yang-Mills connections on principal $G$-bundles, $P$, over arbitrary, closed, Riemannian, smooth manifolds of dimension $d\geq 2$. We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525, arXiv:1510.03815] to remove a positivity constraint on a combination of the Ricci and Riemannian curvatures in a previous $L^{d/2}$-energy gap result due to Gerhardt (2010) and a previous $L^\infty$-energy gap result due to Bourguignon, Lawson, and Simons (1981, 1979), as well as an $L^2$-energy gap result due to Nakajima (1987) for a Yang-Mills connection over the sphere, $S^d$, but with an arbitrary Riemannian metric. The main correction in this version involves replacement of the role of Corollary 4.3 due to Uhlenbeck (1985) and Theorem 5.1 due to the author in the published version of this article at http://dx.doi.org/10.1016/j.aim.2017.03.023 by that of Theorems 1 and 9 due to the author in arXiv:1906.03954.Comment: 26 pages, incorporating final galley proof corrections and corrections to errors noticed since publication online at http://dx.doi.org/10.1016/j.aim.2017.03.023. Background and analytical results provided by arXiv:1409.1525, arXiv:1706.09349, and arXiv:1906.0395

Maximum principles for boundary-degenerate linear parabolic differential operators

We develop weak and strong maximum principles for boundary-degenerate, linear, parabolic, second-order partial differential operators, Lu := -u_t-\tr(aD^2u)-\langle b, Du\rangle + cu, with \emph{partial} Dirichlet boundary conditions. The coefficient, $a(t,x)$, is assumed to vanish along a non-empty open subset, \mydirac_0!\sQ, called the \emph{degenerate boundary portion}, of the parabolic boundary, \mydirac!\sQ, of the domain \sQ\subset\RR^{d+1}, while $a(t,x)$ may be non-zero at points in the \emph{non-degenerate boundary portion}, \mydirac_1!\sQ := \mydirac!\sQ\less\bar{\mydirac_0!\sQ}. Points in \mydirac_0!\sQ play the same role as those in the interior of the domain, \sQ, and only the non-degenerate boundary portion, \mydirac_1!\sQ, is required for boundary comparisons. We also develop comparison principles and a priori maximum principle estimates for solutions to boundary value and obstacle problems defined by boundary-degenerate parabolic operators, again where only the non-degenerate boundary portion, \mydirac_1!\sQ, is required for boundary comparisons. Our results complement those in our previous articles [arXiv1204.6613, arXiv:1305.5098].Comment: 34 pages, 2 figures. This article is the parabolic analogue of arXiv:1204.6613 and restates background material (definitions, notation, spaces) from arXiv:1305.509

On the Morse-Bott property of analytic functions on Banach spaces with Lojasiewicz exponent one half

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Lojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse-Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Lojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space and the Morse Lemma for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Lojasiewicz-Simon gradient inequalities for analytic functions on Banach spaces.Comment: 47 pages, incorporating final galley proof corrections. Background material drawn from arXiv:1708.09775, arXiv:1706.09349, and arXiv:1510.03817v5. To appear in Calculus of Variations and Partial Differential Equation

Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations

We develop strong and weak maximum principles for boundary-degenerate elliptic and parabolic linear second-order partial differential operators, $Au := -\mathrm{tr}(aD^2u)- + cu$, with partial Dirichlet boundary conditions. The coefficient, $a(x)$, is assumed to vanish along a non-empty open subset, $\partial_0\mathscr{O}$, called the \emph{degenerate boundary portion}, of the boundary, $\partial\mathscr{O}$, of the domain $\mathscr{O}\subset\mathbb{R}^d$, while $a(x)$ is non-zero at any point of the \emph{non-degenerate boundary portion}, $\partial_1\mathscr{O} := \partial\mathscr{O}\setminus\overline{\partial_0\mathscr{O}}$. If an $A$-subharmonic function, $u$ in $C^2(\mathscr{O})$ or $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$, is $C^1$ up to $\partial_0\mathscr{O}$ and has a strict local maximum at a point in $\partial_0\mathscr{O}$, we show that $u$ can be perturbed, by the addition of a suitable function $w\in C^2(\mathscr{O})\cap C^1(\mathbb{R}^d)$, to a strictly $A$-subharmonic function $v=u+w$ having a local maximum in the interior of $\mathscr{O}$. Consequently, we obtain strong and weak maximum principles for $A$-subharmonic functions in $C^2(\mathscr{O})$ and $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$ which are $C^1$ up to $\partial_0\mathscr{O}$. Only the non-degenerate boundary portion, $\partial_1\mathscr{O}$, is required for boundary comparisons. Our results extend those in Daskalopoulos and Hamilton (1998), Epstein and Mazzeo [arXiv:1110.0032], and the author [arXiv:1204.6613, 1306.5197], where $\mathrm{tr}(aD^2u)$ is in addition assumed to be continuous up to and vanish along $\partial_0\mathscr{O}$ in order to yield comparable maximum principles for $A$-subharmonic functions in $C^2(\mathscr{O})$, while the results developed here for $A$-subharmonic functions in $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$ are entirely new.Comment: 55 pages, 1 figure, incorporating final galley proof corrections. Includes summary of background material from its companion articles arXiv:1204.6613 and 1306.5197. To appear in Transactions of the American Mathematical Societ

A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps

We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton (1998) in their study of the porous medium equation or the degeneracy of the Heston operator (1993) in mathematical finance. Existence of a solution to the Dirichlet problem on a half-ball, where the operator becomes degenerate on the flat boundary and a Dirichlet condition is only imposed on the spherical boundary, provides the key additional ingredient required for our Perron method. The solution to the Dirichlet problem on the half-ball can be converted to a Dirichlet problem on an infinite slab via a suitable diffeomorphism which becomes holomorphic in dimension two. The required Schauder regularity theory and existence of a solution to the Dirichlet problem on the slab can nevertheless be obtained using previous work of the author and C. Pop [arXiv:1210.6727]. Our Perron method relies on weak and strong maximum principles for degenerate-elliptic operators, concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate-elliptic operators, and maximum and comparison principle estimates previously developed by the author [arXiv:1204.6613].Comment: 61 pages, 5 figure

On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Ito processes

Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that the martingale problem associated with a degenerate-elliptic differential operator with unbounded, locally Holder continuous coefficients on a half-space is well-posed in the sense of Stroock and Varadhan. Second, we prove existence, uniqueness, and the strong Markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable H\"older continuity properties. Third, for an Ito process with degenerate diffusion and unbounded but appropriately regular coefficients, we prove existence of a strong Markov process, unique in the sense of probability law, whose one-dimensional marginal probability distributions match those of the given Ito process.Comment: 27 pages, corresponds to Part 2 of our previous article [arXiv:1112.4824v1]; to appear in Transactions of the American Mathematical Societ

Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions

In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions generalize that of the pure Yang-Mills energy function due to the first author (Theorems 23.1 and 23.17 in arXiv:1409.1525) for base manifolds of dimensions two, three and four and due to Rade (1992) for dimensions two and three.Comment: xvi + 138 pages. To appear in Memoirs of the American Mathematical Societ

Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz proved by Simon (1983). We also prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the functional is Morse-Bott, improving on similar results due to Chill (2003, 2006), Haraux and Jendoubi (2007), and Simon (1996). In our article arXiv:1903.01953, we apply our abstract Lojasiewicz-Simon gradient inequalities to prove a Lojasiewicz-Simon gradient inequalities for the harmonic map energy functional using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities for the harmonic map energy functional generalize those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), and Topping (1997). In our monograph arXiv:1510.03815, we prove Lojasiewicz--Simon gradient inequalities for coupled Yang--Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang--Mills energy function due to the first author (Theorems 23.1 and 23.17 in arXiv:1409.1525) for base manifolds of arbitrary dimension and due to Rade (1992) for dimensions two and three.Comment: 31 pages, incorporating final galley proof corrections. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal