De Giorgi Iteration Research Articles

Maximum principle for non-uniformly parabolic equations and applications

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion and drift coefficients. Moreover, by the Markov selection theorem of Krylov [8], we also establish the existence of the associated strong Markov family.

Multiple Solutions to the Fractional p-Laplacian Equations of Schrödinger–Hardy-Type Involving Concave–Convex Nonlinearities

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the L∞-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for weak solutions to nonlinear elliptic problems involving a singular nonlinear term such as Hardy potentials has not been studied extensively. To overcome this, we utilize a truncated energy technique and the De Giorgi iteration method. As its application, we demonstrate that the problem above has at least two distinct nontrivial solutions by exploiting a variant of Ekeland’s variational principle and the classical mountain pass theorem as the key tools. Furthermore, we prove the existence of a sequence of infinitely many weak solutions that converges to zero in the L∞-norm. To derive this result, we employ the modified functional method and the dual fountain theorem.

New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems

In this paper we present new embedding results for Musielak–Orlicz Sobolev spaces of double phase type. Based on the continuous embedding of W^{1,mathcal {H}}(Omega ) into L^{mathcal {H}_*}(Omega ), where mathcal {H}_* is the Sobolev conjugate function of mathcal {H}, we present much stronger embeddings as known in the literature. Based on these results, we consider generalized double phase problems involving such new type of growth with Dirichlet and nonlinear boundary condition and prove appropriate boundedness results of corresponding weak solutions based on the De Giorgi iteration along with localization arguments.

Local boundedness under nonstandard growth conditions

This paper deals with local boundedness under nonstandard growth conditions by using De Giorgi's iteration method. We first consider quasilinear elliptic systems, and assume that the coefficients satisfying some boundedness and coercivity conditions, then we derive local boundedness of solutions. Secondly, we consider elliptic equations with a drift term, under some nonstandard growth conditions, we derive that solutions are locally bounded. Finally, we consider variational integrals with the integrand satisfy nonstandard growth conditions, local boundedness of minimizers is derived.

Nondegeneracy for stable solutions to the one-phase free boundary problem

We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for the mean curvature of the free boundary. We then apply the nondegeneracy estimate to obtain local curvature bounds for stable free boundaries in dimension n, provided the Bernstein-type theorem for stable, entire solutions in the same dimension is valid. In particular, we obtain this curvature estimate in $$n=2$$ dimensions.

Well-posedness of nonlinear two-phase flow model with solute transport

In this paper, we consider the nonlinear two-phase flow problem with solute transport, in which an imperfect coupling condition is imposed on the phase interface. The interface condition here means that the jump of the solution at the interface is proportional to the normal flux. A weak dissipativity condition is imposed for the reaction term, and the well-posedness of the weak solution for the nonlinear problem is proved in the conventional solution space. The proof is based on a specific functional analytic framework that uses the De Giorgi iteration and the fixed point theorem.

A-priori bound and Hölder continuity of solutions to degenerate elliptic equations with variable exponents

We investigate the boundedness and regularity of solutions to degenerate elliptic equations with variable exponents that are subject to the Dirichlet boundary condition. By employing the De Giorgi iteration, we obtain a-priori bounds and the Hölder continuity for solutions. As an application, we obtain the existence of infinitely many small solutions for a class of degenerate elliptic equations involving variable exponents.

Relative Stability in the Sup-Norm and Input-to-State Stability in the Spatial Sup-Norm for Parabolic PDEs

In this article, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">relative <inline-formula><tex-math notation="LaTeX">$\mathcal {K}$</tex-math></inline-formula>-equi-stability</i> (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic partial differential equations (PDEs). Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. An example concerned with a superlinear parabolic PDE with Robin boundary condition is provided to illustrate the obtained ISS results. Besides, as an application of the notion of RKES, we conduct stability analysis for a class of parabolic PDEs in cascade coupled over the domain or on the boundary of the domain, in the spatial and time sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi iteration is extensively used in the proof of the results presented in this article.

On the logarithmic type boundary modulus of continuity for the Stefan problem: To the memory of Emmanuele DiBenedetto

A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain satisfies a measure density property, and the boundary datum has a logarithmic type modulus of continuity. For the Neumann problem, we assume that the lateral boundary is smooth, and the boundary datum is bounded. The proofs are measure theoretical in nature, exploiting De Giorgi's iteration and refining DiBenedetto's approach. Based on the sharp quantitative estimates, construction of continuous weak (physical) solutions is also indicated. The logarithmic type modulus of continuity has been conjectured to be optimal as a structural property for weak solutions to such partial differential equations.

Formation of singularities for multi-dimensional transport equations with nonlocal velocity

This paper is concerned with a class of multi-dimensional transport equations with nonlocal velocity. It is shown that the local smooth solution cannot exist globally in time via the De Giorgi iteration technique.

Existence and multiplicity of solutions to concave–convex-type double-phase problems with variable exponent

This paper is devoted to the study of the L∞-bound of solutions to the double-phase nonlinear problem with variable exponent by the case of a combined effect of concave–convex nonlinearities. The main tools are the De Giorgi iteration method and a truncated energy technique. Applying this and a variant of Ekeland’s variational principle, we give the existence of at least two distinct nontrivial solutions belonging to L∞-space when the condition on a nonlinear convex term does not assume the Ambrosetti–Rabinowitz condition in general. In addition, our problem admits a sequence of small energy solutions whose converge to zero in L∞ space. To achieve this result, we apply the modified functional method and global variational formulation as the main tools.

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao (1999) [12] and Winkert-Zacher (2012) [47] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the C1,α-regularity follows immediately.

On the Hölder regularity of signed solutions to a doubly nonlinear equation

We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is∂t(|u|p−2u)−Δpu=0,p>1. The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the Hölder regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain, up to the parabolic boundary. In addition, based on the expansion of positivity, we are able to give an alternative proof of Harnack's inequality for non-negative solutions. Moreover, as a consequence of the interior estimates, we also obtain a Liouville-type result.

Second Derivatives Estimate of Suitable Solutions to the 3D Navier–Stokes Equations

We study the second spatial derivatives of suitable weak solutions to the incompressible Navier-Stokes equations in dimension three. We show that it is locally $L ^{\frac43, q}$ for any $q > \frac43$, which improves from the current result $L ^{\frac43, \infty}$. Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations.

On a class of $p(x)$-Laplacian equations without any growth and Ambrosetti-Rabinowitz conditions

The aim of this paper is to establish the existence of nontrivial solutions for $p(x)$-Laplacian equations without any growth and Ambrosetti-Rabinowitz conditions. Employing the cutoff function approach, we show that auxiliary problem has at least one nontrivial solution. Furthermore, we obtain nontrivial solutions for original problems using De Giorgi iteration. The results presented here extend some recent contributions obtained for problems driven by the $p(x)$-Laplacian or even to more general differential operators.

Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations

We prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w.r.t. time belongs to the large class of PC kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the De Giorgi iteration technique. Sufficiently regular solutions are shown to be unique by means of an L1-contraction estimate.

MULTIPLICITY OF SOLUTIONS FOR DOUBLE PHASE EQUATIONS WITH CONCAVE-CONVEX NONLINEARITIES

This paper is devoted to the study of the L ∞-bound of solutions to a double-phase problem with concave-convex nonlinearities by applying the De Giorgi's iteration method and the localization method. Employing this and a variant of Ekeland's variational principle, we provide the existence of at least two distinct nontrivial solutions belonging to L ∞-space when the convex term does not satisfy the Ambrosetti-Rabinowitz condition in general. In addition, our problem has a sequence of multiple small energy solutions whose L ∞-norms converge to zero. To achieve this result, we utilize the modified functional method and the dual fountain theorem as the main tools.

The Dirichlet problem for nonlocal elliptic equations

We study a class of nonlocal elliptic equations on a bounded domain . For (), by the Lax–Milgram theorem and the De Giorgi iteration, we prove the existence and uniqueness of solution. Furthermore, we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations.

Solutions of the 4-species quadratic reaction-diffusion system are bounded and C∞-smooth, in any space dimension

We establish the boundedness of solutions of reaction-diffusion systems with qua-dratic (in fact slightly super-quadratic) reaction terms that satisfy a natural entropy dissipation property, in any space dimension N > 2. This bound imply the C ∞-regularity of the solutions. This result extends the theory which was restricted to the two-dimensional case. The proof heavily uses De Giorgi's iteration scheme, which allows us to obtain local estimates. The arguments relies on duality reasonings in order to obtain new estimates on the total mass of the system, both in L (N +1)/N norm and in a suitable weak norm. The latter uses C α regularization properties for parabolic equations.

Formula omitted]-theory of linear degenerate SPDEs and [formula omitted] ([formula omitted]) estimates for the uniform norm of weak solutions

In this paper, we are concerned with possibly degenerate stochastic partial differential equations (SPDEs). An L2-theory is introduced, from which we derive a Hörmander-type theorem with an analytical approach. With the method of De Giorgi iteration, we obtain the maximum principle which states the Lp (p>0) estimates for the time-space uniform norm of weak solutions.