Parabolic Differential Operator Research Articles

On the uniqueness theorems for transmission problems related to models of elasticity, diffusion and electrocardiography

Abstract We consider a generalization of the inverse problem of the electrocardiography in the framework of the theory of elliptic and parabolic differential operators. More precisely, starting with the standard bidomain mathematical model related to the problem of the reconstruction of the transmembrane potential in the myocardium from known body surface potentials, we formulate a more general transmission problem for elliptic and parabolic equations in the Sobolev type spaces and describe conditions, providing uniqueness theorems for its solutions. Next, the new transmission problem is interpreted in the framework of the elasticity theory applied to composite media. Finally, we prove a uniqueness theorem for an evolutionary transmission problem that can be easily adopted to many models involving the diffusion type equations.

Parabolic opers and differential operators

Parabolic SL(r,C)-opers were defined and investigated in [BDP] in the set-up of vector bundles on curves with a parabolic structure over a divisor. Here we introduce and study holomorphic differential operators between parabolic vector bundles over curves. We consider the parabolic SL(r,C)-opers on a Riemann surface X with given singular divisor S and with fixed parabolic weights satisfying the condition that all parabolic weights at any point $x_i$ in S are integral multiples of $\frac{1}{2N_i+1}$, where $N_i > 1$ are fixed integers. We prove that this space of opers is canonically identified with the affine space of holomorphic differential operators of order r between two natural parabolic line bundles on X (depending only on the divisor S and the weights $N_i$) satisfying the conditions that the principal symbol of the differential operators is the constant function 1 and the sub-principal symbol vanishes identically. The vanishing of the sub-principal symbol ensures that the logarithmic connection on the rank r bundle is actually a logarithmic SL(r, C)-connection.

Comparison principles for second-order elliptic/parabolic equations with discontinuities in the gradient compatible with Finsler norms

This paper is about elliptic and parabolic partial differential operators with discontinuities in the gradient which are compatible with a Finsler norm in a sense to be made precise. Examples of this type of problems arise in a number of contexts, most notably the recent work of Chatterjee and the second author [7] on scaling limits of discrete surface growth models as well as L∞−variational problems. Building on the approach of Ishii [16], new comparison results are proven within a unified framework that includes a number of previous results as special cases.

An open mapping theorem for nonlinear operator equations associated with elliptic complexes

Let be the elliptic complex on an n-dimensional smooth closed Riemannian manifold X with the first-order differential operators and smooth vector bundles over X. We consider nonlinear operator equations, associated with the parabolic differential operators , generated by the Laplacians of the complex , in special Bochner–Sobolev functional spaces. We prove that under reasonable assumptions regarding the nonlinear term the Frechét derivative of the induced nonlinear mapping is continuously invertible and the map is open and injective in chosen spaces.

An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators

For a constant coefficient partial differential operator P(D) with a single characteristic direction, such as the time-dependent free Schrödinger operator as well as non-degenerate parabolic differential operators like the heat operator, we characterize when open subsets X1⊆X2 of Rd form a P-Runge pair. The presented condition does not require any kind of regularity of the boundaries of X1 nor X2. As part of our result we prove that for a large class of non-elliptic operators P(D) there are smooth solutions u of the equation P(D)u=0 on Rd with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for P(D).

On Initial Boundary Value Problem for Parabolic Differential Operator with Non-coercive Boundary Conditions

We consider initial boundary value problem for uniformly 2-parabolic differential operator of second order in cylinder domain in Rn with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo-Galerkin method we prove that problem has unique solution in special Bochner space

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, A u := − t r ( a D 2 u ) − ⟨ b , D u ⟩ + c u Au := -\mathrm {tr}(aD^2u)-\langle b, Du\rangle + cu , with partial Dirichlet boundary conditions. The coefficient, a ( x ) a(x) , is assumed to vanish along a nonempty open subset, ∂ 0 O \partial _0\mathscr {O} , called the degenerate boundary portion, of the boundary, ∂ O \partial \mathscr {O} , of the domain O ⊂ R d \mathscr {O}\subset \mathbb {R}^d , while a ( x ) a(x) is nonzero at any point of the nondegenerate boundary portion, ∂ 1 O := ∂ O ∖ ∂ 0 O ¯ \partial _1\mathscr {O} := \partial \mathscr {O}\setminus \overline {\partial _0\mathscr {O}} . If an A A -subharmonic function, u u in C 2 ( O ) C^2(\mathscr {O}) or W l o c 2 , d ( O ) W^{2,d}_{\mathrm {loc}}(\mathscr {O}) , is C 1 C^1 up to ∂ 0 O \partial _0\mathscr {O} and has a strict local maximum at a point in ∂ 0 O \partial _0\mathscr {O} , we show that u u can be perturbed, by the addition of a suitable function w ∈ C 2 ( O ) ∩ C 1 ( R d ) w\in C^2(\mathscr {O})\cap C^1(\mathbb {R}^d) , to a strictly A A -subharmonic function v = u + w v=u+w having a local maximum in the interior of O \mathscr {O} . Consequently, we obtain strong and weak maximum principles for A A -subharmonic functions in C 2 ( O ) C^2(\mathscr {O}) and W l o c 2 , d ( O ) W^{2,d}_{\mathrm {loc}}(\mathscr {O}) which are C 1 C^1 up to ∂ 0 O \partial _0\mathscr {O} . Points in ∂ 0 O \partial _0\mathscr {O} play the same role as those in the interior of the domain, O \mathscr {O} , and only the nondegenerate boundary portion, ∂ 1 O \partial _1\mathscr {O} , is required for boundary comparisons. Moreover, we obtain a comparison principle for a solution and supersolution in W l o c 2 , d ( O ) W^{2,d}_{\mathrm {loc}}(\mathscr {O}) to a unilateral obstacle problem defined by A A , again where only the nondegenerate boundary portion, ∂ 1 O \partial _1\mathscr {O} , is required for boundary comparisons. Our results extend those of Daskalopoulos and Hamilton, Epstein and Mazzeo, and Feehan, where t r ( a D 2 u ) \mathrm {tr}(aD^2u) is in addition assumed to be continuous up to and vanish along ∂ 0 O \partial _0\mathscr {O} in order to yield comparable maximum principles for A A -subharmonic functions in C 2 ( O ) C^2(\mathscr {O}) , while the results developed here for A A -subharmonic functions in W l o c 2 , d ( O ) W^{2,d}_{\mathrm {loc}}(\mathscr {O}) are entirely new. Finally, we obtain analogues of all the preceding results for parabolic linear second-order partial differential operators, L u := − u t − t r ( a D 2 u ) − ⟨ b , D u ⟩ + c u Lu := -u_t - \mathrm {tr}(aD^2u)-\langle b, Du\rangle + cu .

Existence, iteration procedures and directional differentiability for parabolic QVIs

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

Regular Boundary Points and Exit Distributions for Parabolic Differential Operators

We show that the regularity of a boundary point for a parabolic differential operator in divergence form is under some geometric assumptions equivalent to the property that the density of the exit distribution for a time reversed process vanishes at that point. We give regularity and irregularity criterions for equations with variable coefficients. Thus, the known result on the Fulks measure that states that the density with respect to the Lebesgue measure vanishes at the point opposite to the center of the heat ball (see Fulks (Proc. Am. Math. Soc. 17, 6–11 1966)) can be extended to exit distributions for more general regions and parabolic differential operators.

On the fundamental solution of the Fokker–Planck–Kolmogorov equation

The Fokker–Planck–Kolmogorov parabolic second-order differential operator is considered, for which its fundamental solution is derived in explicit form. Such operators arise in numerous applications, including signal filtering, portfolio control in financial mathematics, plasma physics, and problems involving linear-quadratic regulators.

Hodge-type Decomposition for Time-dependent First-order Parabolic Operators with Non-constant Coefficients: The Variable Exponent Case

In this paper we present a Hodge-type decomposition for variable exponent spaces. More concretely, we address some time-dependent parabolic firstorder partial differential operators with non-constant coefficients, where one of the components is the kernel of the parabolic-type Dirac operator. This decomposition is presented over different types of domains in the n-dimensional Euclidean space n-dimensional Euclidean space \({\mathbb{R}^{n}}\). The case of the time-dependent Schrodinger operator is included as a special case within this context.

On Asymptotic Effects of Boundary Perturbations in Exponentially Shaped Josephson Junctions

A parabolic integro differential operator \(\mathcal{L}\), suitable to describe many phenomena in various physical fields, is considered. By means of equivalence between \(\mathcal{L}\) and the third order equation describing the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, explicitly evaluating, boundary contributions related to the Dirichlet problem.

Continuous maximal regularity on uniformly regular Riemannian manifolds

We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on uniformly regular Riemannian manifolds M. As an application, we show that solutions to the Yamabe flow on M instantaneously regularize and become real analytic in space and time. The regularity result is obtained by introducing a family of parameter-dependent diffeomorphisms acting on functions on M in conjunction with maximal regularity and the implicit function theorem.

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.

On the convergence of projected triangular decomposition methods for pricing American options with stochastic volatility

Numerical pricing of American options with Heston stochastic volatility model is considered. The complementarity problem with a two-dimensional parabolic partial differential operator is discretized by the Craig–Sneyd alternative direction implicit scheme, and the resulted linear complementarity problems at each time step are solved by the projected triangular decomposition methods, which are constructed as an extension of the classical Brennan Schwartz algorithm. The convergence theorems are established when the system matrix is an M-matrix. Numerical experiments show that the proposed methods with alternative direction implicit schemes are efficient and outperform the classical PSOR method and operator splitting method.

Hodge decomposition and solution formulas for some first-order time-dependent parabolic operators with non-constant coefficients

In this paper, we present a Hodge decomposition for the \(L_p\)-space of some parabolic first-order partial differential operators with non-constant coefficients. This is done over different types of domains in Euclidean space \(\mathbb{R }^n\) and on some conformally flat cylinders and the \(n\)-torus associated with different spinor bundles. Initially, we apply a regularization procedure in order to control the non-removable singularities over the hyperplane \(t=0\). Using the setting of Clifford algebras combined with a Witt basis, we introduce some specific integral and projection operators. We present an \(L_p\)-decomposition where one of the components is the kernel of the regularized parabolic Dirac operator with non-constant coefficients. After that, we study the behavior of the solutions and the validity of our results when the regularization parameter tends to zero. To round off, we give some analytic solution formulas for the special context of domains on cylinders and \(n\)-tori.

Existence and uniqueness for a nonlinear inverse reaction‐diffusion problem with a nonlinear source in higher dimensions

This paper analyzes the existence and the uniqueness problem for an n‐dimensional nonlinear inverse reaction‐diffusion problem with a nonlinear source. A transformation is used to obtain a new inverse coefficient problem. Then, a parabolic differential operator Lλ is defined to establish the relation between the solution of Lλ = 0 and the new inverse problem. Following this, it is shown that the inverse problem has at least one solution in the class of admissible coefficients. Furthermore, it is proved that this solution is the unique solution of the undertaken inverse problem. A numerical example is given to illustrate ill‐posedness of the inverse problem. Copyright © 2013 John Wiley & Sons, Ltd.

Safe-Parking Framework for Fault-Tolerant Control of Transport−Reaction Processes

This work considers the problem of handling actuator faults in transport−reaction processes described by quasi-linear parabolic partial differential equations (PDEs) subject to input constraints. To this end, first, by exploiting the separation between the fast and slow eigenmodes of the parabolic spatial differential operator in combination with Galerkin’s method, a finite set of ordinary differential equations (ODEs) that captures the dominant dynamics of the PDE system are constructed. This finite ODE system is used to develop a Lyapunov-based model predictive controller which provides an explicit characterization of the set of initial conditions from where closed-loop stability of the parabolic PDE system is guaranteed. This control design is then subsequently used to develop a safe-parking framework for handling faults. In particular, faults which preclude the possibility of maintaining operation at the nominal equilibrium distribution, using the existing robust or reconfiguration-based fault-tolerant control approaches are considered. The key idea is to temporarily maintain the process at an appropriate “safe-park” distribution using the available depleted control action. This “safe-park” distribution is chosen to prevent onset of hazardous situations as well as to ensure smooth resumption of nominal operation upon fault repair. Utilizing the stability region characterization provided by the developed predictive controller, safe-park distributions from the safe-park candidates (equilibrium distributions subject to the remaining functioning and fail-safe values of the failed actuators) are chosen to preserve closed-loop stability upon fault repair. The proposed framework is illustrated on a diffusion−reaction process.

Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type

Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.

Operator splitting methods for pricing American options under stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.