Approximation of the Solutions to Quasilinear Parabolic Problems with Perturbed $$VMO_x$$ Coefficients

The obstacle problem for parabolic non-divergence form operators of Hörmander type

We investigate the Cauchy problem for second-order hyperbolic operators in the framework of the space of C∞ functions. In the case where the coefficients of their principal parts depend only on the time variable and are real analytic, we give a sufficient condition for C∞ well-posedness, which is also a necessary one when the space dimension is less than 3 or the coefficients of the principal parts are semi-algebraic functions (e.g., polynomials) of the time variable.

Approximation results for bilateral nonlinear problems of nonvariational type

We prove the existence of unique solutions to Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies an exterior measure condition.

We study regularity properties of weak solutions in the Sobolev space \({W^{1,n}_0}\) to inhomogeneous elliptic systems under a natural growth condition and on bounded Lipschitz domains in \({\mathbb{R}^n}\) , i. e. we investigate weak solutions in the limiting situation of the Sobolev embedding. Several counterexamples of irregular solutions are constructed in cases, where additional structure conditions might have led to regularity. Among others we present both bounded irregular and unbounded weak solutions to elliptic systems obeying a one-sided condition, and we further construct unbounded extremals of two-dimensional variational problems. These counterexamples do not exclude the existence of a regular solution. In fact, we establish the existence of regular solutions—under standard assumptions on the principal part and the aforementioned one-sided condition on the inhomogeneity. This extends previous works for n = 2 to more general cases, including arbitrary dimensions. Moreover, this result is achieved by a simplified proof invoking modern techniques.

We deal with an inverse problem of determining a coefficient a(x, t) of principal part for second order parabolic equations with non-divergent form when the solution is known. Such a problem has important applications in a large fields of applied science. We propose a well-posed approximate algorithm to identify the coefficient. The existence, uniqueness and stability of such solutions a(x, t) are proved. A necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. Our numerical simulations show that the coefficient is recovered very well.

In the paper, we prove the existence, uniqueness and differentiable dependence of solutions for some nonlinear Urysohn integral equations on parameters. Some sufficient conditions for the nonlinear integral operator of the Urysohn type to be a diffeomorphism are stated. Global invertibility of the Urysohn operator in a certain Sobolev space is ascertained. Consequently, global solvability of Urysohn equations is claimed. Similar results are obtained for some nonlinear Urysohn integral equations with controls by the use of the global implicit function theorem published in the recent paper by Idczak. The proofs of global diffeomorphisms and global implicit functions theorems, the main tools used in the paper, rely in an essential way on the mountain pass theorem. Applications of results to some specific nonlinear Urysohn integral equations are also presented.

We prove a boundary Harnack inequality for nonlocal elliptic operators L in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if Lu1 = Lu2 = 0 in Ω ∩ B1, u1 = u2 = 0 in B1 ∖Ω, and u1,u2 ≥ 0 in ℝn, then u1 and u2 are comparable in B1/2. The result applies to arbitrary open sets Ω. When Ω is Lipschitz, we show that the quotient u1/u2 is Holder continuous up to the boundary in B1/2. These results will be used in forthcoming works on obstacle-type problems for nonlocal operators.

We consider semilinear second-order parabolic equations whose principal parts may have either divergence or nondivergence form and whose nonlinear terms satisfy conditions of Bernstein-Dini type. We study the qualitative properties of the classical solutions of nondivergence equations and generalized solutions of equations with divergent principal parts: the behavior of solutions in various unbounded domains and near the boundaries of domains, removability of singularities of solutions, vanishing of solutions in unbounded domains, in particular solutions of compact support and uniqueness and continuous dependence on the boundary conditions for solutions of the exterior initial/boundary problem.

Chapter 6 Nonlinear spectral problems for degenerate elliptic operators

Presents modern methods and techniques for solving boundary value problems for nonlinear elliptic operators. Focus is upon existence and bifurcation results in appropriate Sobolev spaces. The subject of this book is the theory of nonlinear boundary value problems for elliptic operators with degeneration and singularity. It focuses on the existence and bifurcation of weak solutions in appropriate weighted Sobolev spaces. The main tools are functional analytic methods based on the variational approach and the theory of topological degree of monotone type nonlinear mapping. Topics covered include: existence results for higher order boundary value problems in a rather general setting. An extensive study of the p-Laplacian and its degenerated and singular modifications, mainly existence and bifurcation results on bounded domains as well as on the whole Euclidean space. The text requires some basic knowledge of nonlinear functional analysis and differential equations. Elementary facts on function spaces and the theory of nonlinear operators are discussed in the first chapter.

We consider the existence of solutions of variational inequality form. Findu∈D(J):〈A(u),v-u〉+〈F(u),v-u〉+J(v)-J(u)≥0,∀v∈W1LM(Ω),whose principal part is having a growth not necessarily of polynomial type, whereAis a second-order elliptic operator of Leray-Lions type,Fis a multivalued lower order term, andJis a convex functional. We use subsupersolution methods to study the existence and enclosure of solutions in Orlicz-Sobolev spaces.

The solvability in Sobolev spaces W 1,2 p is proved for nondivergence form second-order parabolic equations for p > 2 close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the W 2 p -solvability for the same p of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when p = 2.

We prove the unique solvability of second order elliptic equations in nondivergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the Martingale problem associated with the elliptic equations.

We consider time fractional parabolic equations in divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$, which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, and on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.