New harmonic self-maps constructed for specific Lie groups.
problem Harmonic self-maps of compact cohomogeneity one manifolds.
method Developed theory of equivariant harmonic self-maps; constructed new maps by solving non-linear boundary value problems.
result Harmonic self-maps of specific Lie groups with novel degrees.
The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.
problem Boundary estimates for solutions to fully non-linear elliptic equations on Hermitian manifolds.
method Unified approach using quantitative boundary estimates, gradient estimates, and existence results.
result Established gradient estimates and unified approach to Dirichlet problem solutions.
We present a stochastic numerical method for solving fully non-linear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the non-linearity.
Solves Dirichlet problem for elliptic equations on Hermitian manifolds.
problem Solving Dirichlet problem for fully non-linear elliptic equations on Hermitian manifolds.
method Establishing a quantitative boundary estimate under a subsolution assumption.
result Derives solvability and regularity of the Dirichlet problem.
We consider that the price of a firm follows a non linear stochastic delay differential equation. We also assume that any claim value whose value depends on firm value and time follows a non linear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a Random Partia…
In this paper, we investigate the non-linear Black--Scholes equation: ut+ax2uxx+bx3uxx2+c(xux−u)=0,a,b>0, c≥0. and show that the one can be reduced to the equation ut+(uxx+ux)2=0 by an appropriate point transformation of variables. For the resulting equation, we study the group-theore…
Surveying inverse problems on manifolds with boundaries.
problem Inverse problems for geometric structures on manifolds with boundaries.
method Analyzes linear and non-linear geometric inverse problems.
result Recent results on geodesic X-ray transforms and Riemannian metrics.
Develops deep learning methods for non-linear PDEs in credit risk.
problem Solving option XVA pricing problems with non-linear PDE models.
method Boundary-safe PINNs approach, using automatic differentiation.
result Eliminates heuristic boundary condition weights, improves accuracy.
New solutions found for complex-valued harmonic morphisms with rational exponents.
problem Finding new solutions to complex-valued harmonic morphisms.
method Generalizing from polynomial solutions to rational exponents.
result Many new solutions to the problem.
We study the problem of inviscid slightly compressible fluids in a bounded domain. We find a unique solution to the initial-boundary value problem and show that it is near the analogous solution for an incompressible fluid provided the initial conditions for the two problems are close. In particular, the divergence of …
We study a multiplicative transient price impact model for an illiquid financial market, where trading causes price impact which is multiplicative in relation to the current price, transient over time with finite rate of resilience, and non-linear in the order size. We construct explicit solutions for the optimal contr…
Study shows X-ray transform injectivity for hyperbolic manifolds.
problem Recovering metrics from boundary measurements on hyperbolic manifolds.
method Injectivity of X-ray transform in several cases.
result Injectivity of X-ray transform proven for asymptotically hyperbolic manifolds.
We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donalds…
From the Hamilton-Jacobi-Bellman equation for the value function we derive a non-linear partial differential equation for the optimal portfolio strategy (the dynamic control). The equation is general in the sense that it does not depend on the terminal utility and provides additional analytical insight for some optimal…
Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.
problem Solving elliptic equations on Hermitian manifolds with boundary conditions.
method Derives quantitative boundary estimates and proves existence of solutions.
result Proves existence of solutions under almost optimal structural conditions.
Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.
Let (Mn,g), n≥3 be a noncompact complete Riemannian manifold with compact boundary and f a smooth function on ∂M. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of g that is complete, has zero scalar curvature on M and has mean curv…
Paper tackles robust control of SDEs with ambiguity, proving value function existence and applying to investment problems.
problem Robust control of SDEs with ambiguity parameters and non-Lipschitz coefficients.
method Existence and uniqueness of value function established through BSDEs with non-linear growth conditions.
result Existence and uniqueness of value function in proper space, verified through BSDEs.
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The boundary conditions in this theory are taken as elements of the C^*-algebra generated by p…
Study of Hamiltonian boundary value problems with symplectic symmetries.
problem Bifurcations of solutions in Hamiltonian boundary value problems.
method Convenient, coordinate-free framework for separated Lagrangian boundary value problems; analysis of conformal symplectic symmetries.
result Existence of obstructions due to conformal symplectic symmetries on bifurcation behavior.
This thesis constructs cmc 1/2 surfaces from catenoids, proving convergence and solving boundary value problems.
problem Creating cmc 1/2 surfaces with positive genus in H2imesR. method Analytic gluing construction, solving mean curvature equation via perturbative methods and linear analysis.
result Construction of cmc 1/2 annuli asymptotic to horizontal catenoids, proving convergence to horocylinders.
We reveal an interesting convex duality relationship between two problems: (a) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (b) a controller-and-stopper problem, in which the controller controls the drift a…
Study boundary value problems for elliptic operators on manifolds.
problem Characterize and analyze boundary conditions for first-order elliptic differential operators.
method Develops a new framework for elliptic boundary conditions, proving equivalence and regularity of solutions.
result Elliptic boundary conditions yield a Fredholm operator on compact manifolds.
Moving boundary problems allow to model systems with phase transition at an inner boundary. Driven by problems in economics and finance, in particular modeling of limit order books, we consider a stochastic and non-linear extension of the classical Stefan-problem in one space dimension, where the paths of the moving in…
New method reconstructs strain and lattice spacing from neutron data.
problem Jointly reconstructing strain and lattice spacing from neutron data.
method Solves non-linear problem ensuring strain field equilibrium with knowledge of boundary conditions.
result Demonstrates ability to jointly reconstruct strain and lattice spacing from simulated data.
Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…
Elliptic boundary value problem for G2 structures on manifolds.
problem Solving G2 holonomy equation on manifolds with boundary.
method Setting up a suitable linear elliptic boundary value problem.
result Existence of certain G2 cobordisms between deformations of Calabi-Yau 3-folds.
Study of boundary value problems in 7, 4, and 3 dimensions with G2 and holonomy structures.
problem Boundary value problems for G2 structures in 7-manifolds with prescribed 3-forms.
method General observations and detailed study of reductions to 4 and 3 dimensions by imposing symmetry.
result 3-dimensional reduction admits a dual variational formulation using the real Monge-Ampere equation.
Paper connects minimal and maximal surfaces' boundary value problems.
problem Existence and uniqueness of minimal surfaces' boundary value problems.
method One-to-one correspondence between minimal and maximal surfaces' solutions.
result One-to-one correspondence between minimal and maximal surfaces' boundary value problems.
A guide for solving first-order elliptic boundary value problems.
problem Solving first-order elliptic boundary value problems on manifolds.
method Operator methods and general elliptic boundary conditions.
result Characterization of a new subclass of elliptic boundary conditions.
Method proves ellipticity of vacuum spacetime boundary problems.
problem Proving ellipticity of boundary value problems for stationary vacuum spacetimes.
method Developed a general method using the projection formalism.
result Proved manifold theorem for moduli space of stationary vacuum spacetimes.
We study boundary value problems for the Dirac operator on Riemannian Spinc manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …
Formula for Z_2-valued index of symmetric operators on manifolds.
problem Index of symmetric operators on manifolds with boundary.
method Cohomological formula for Z_2-valued index.
result A formula for the Z_2-valued index of operators on manifolds.
New solutions found using rational exponents in complex-valued geometry.
problem Difficult non-linear problems in differential geometry.
method Generalizing from polynomial solutions to rational exponents.
result Many new solutions found.
The subject of this paper is an optimal consumption/optimal portfolio problem with transaction costs and with multiple risky assets. In our model the transaction costs take a special form in that transaction costs on purchases of one of the risky assets (the endowed asset) are infinite, and transaction costs involving …
We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.
Uniform Shapiro-Lopatinski conditions ensure well-posedness of boundary value problems on manifolds with bounded geometry.
problem Boundary value problems on manifolds with boundary and bounded geometry.
method Uniform Shapiro-Lopatinski regularity condition, compactness argument, Nirenberg trick.
result Uniform Shapiro-Lopatinski condition characterizes well-posed boundary value problems.
New Shapley values reveal non-linear feature dependencies.
problem Understanding non-linear dependencies in machine learning models.
method Model-independent Shapley values using non-parametric measures of dependence.
result Model-independent Shapley values can uncover non-linear dependencies.
Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space H1. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Mar…
We discuss the ζ−regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.
New boundary conditions improve Hamiltonian analysis in GR.
problem Improving Hamiltonian analysis in GR with IBVP.
method Presented and analyzed new boundary conditions.
result New boundary conditions lead to better Hamiltonian analysis.
Solves initial boundary value problem for vacuum Einstein equations and proves geometric uniqueness.
problem Initial boundary value problem for vacuum Einstein equations.
method Formulated IBVP, solved simultaneously in local harmonic coordinates, constructed unique maximal globally hyperbolic solution.
result Vacuum spacetimes satisfying fixed initial-boundary conditions and corner conditions are geometrically unique near the initial surface.
Study calculates index for non-compact manifolds with boundary.
problem Index of strongly Callias-type operators on non-compact manifolds.
method Computes index of local boundary value problem on non-compact manifold.
result Extends Freed's index theorem to non-compact manifolds.
Paper solves a mixed boundary value problem in space forms with umbilical boundaries.
problem Solving a partially overdetermined mixed boundary value problem in space forms.
method Generalizing previous results to domains with partial umbilical boundaries.
result A partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface.
C-ADAM is a new adaptive solver for complex nested problems.
problem Solving compositional problems involving nested expected values.
method Adaptive solver for non-linear functional nesting of expected values.
result C-ADAM converges to a stationary point in O(δ−2.25). Study of elliptic boundary value problems on non-compact manifolds.
problem Analyzing elliptic differential operators on manifolds with non-compact boundaries.
method Regularity theory and trace theorems for sections in the maximal domain under various assumptions.
result Systematic study of local and nonlocal boundary conditions, including the Atiyah-Patodi-Singer condition.
Develops a method to forecast non-linear time series.
problem Inability of traditional methods to handle non-linear dependencies in non-Gaussian series.
method Learning vector-valued functions in reproducing kernel Hilbert space, learning multiple matrix-valued kernels.
result Superior predictive performance and recovery of dynamic relationships.