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.     

Regularity of Weak Solutions and Their Attractors for a Parabolic Feedback Control Problem

We investigate additional regularity properties of all globally defined weak solutions, their global and trajectory attractors for a class of autonomous differential inclusion with upper semi-continuous interaction function, when initial data $u_{\tau}\in L^2(\Omega)$ . The main contributions of this paper are: (i) additional regularity and new topological properties of all weak solutions of parabolic feedback control problem with upper semi-continuous interaction function, (ii) a sufficient condition for regularity of global and trajectory attractors, and (iii) new a priory estimates for all weak solutions.     

Higher regularity of the free boundary in the parabolic Signorini problem

We show that the quotient of two caloric functions which vanish on a portion of an \(H^{k+ \alpha }\) regular slit is \(H^{k+ \alpha }\) at the slit, for \(k \ge 2\). In the case \(k=1\), we show that the quotient is in \(H^{1+\alpha }\) if the slit is assumed to be space-time \(C^{1, \alpha }\) regular. This can be thought of as a parabolic analogue of a recent important result in De Silva and Savin (Boundary Harnack estimates in slit domains and applications to thin free boundary problems, 2014), whose ideas inspired us. As an application, we show that the free boundary near a regular point of the parabolic thin obstacle problem studied in Danielli et al. (Optimal regularity and the free boundary in the parabolic Signorini problem. Mem. Am. Math. Soc., 2013) with zero obstacle is \(C^{\infty }\) regular in space and time.     

Regularity results for fully nonlinear parabolic integro-differential operators

In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov–Backelman–Pucci estimate with $0<\sigma <2$ . And we show a Harnack inequality, Hölder regularity, and $C^{1,\alpha }$ -regularity of the solutions by obtaining decay estimates of their level sets.     

Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ?², where ? takes arbitrary values in the interval (0, 1]. When ? vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ? tends to zero, a parabolic boundary layer with a characteristic width ? appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 , where \(N = \mathop {\min }\limits_s N_s \), N ^s + 1 and N ₀ + 1 are the numbers of mesh points on the axes x _s and t, respectively.     

A Family of Functions with Two Different Spectra of Singularities

Our goal is to study the multifractal properties of functions of a given family which have few non vanishing wavelet coefficients. We compute at each point the pointwise Hölder exponent of these functions and also their local \(L^p\) regularity, computing the so-called \(p\) -exponent. We prove that in the general case the Hölder and \(p\) -exponent are different at each point. We also compute the dimension of the sets where the functions have a given pointwise regularity and prove that these functions are multifractal both from the point of view of Hölder and \(L^p\) local regularity with different spectra of singularities. Furthermore, we check that multifractal formalism type formulas hold for functions in that family.     

A general regularity theory for weak mean curvature flow

We give a new proof of Brakke’s partial regularity theorem up to $C^{1,\varsigma }$ for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard’s regularity theorem in the sense that the special time-independent case reduces to Allard’s theorem.     

Parabolic equations with dynamical boundary conditions and source terms on interfaces

We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable \(L^p\) -setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.     

Estimates for a spectral parameter in elliptic boundary value problems with discontinuous nonlinearities

Under study are the two classes of elliptic spectral problems with homogeneous Dirichlet conditions and discontinuous nonlinearities (the parameter occurs in the nonlinearity multiplicatively). In the former case the nonlinearity is nonnegative and vanishes for the values of the phase variable not exceeding some positive number c; it has linear growth at infinity in the phase variable u and the only discontinuity at u = c. We prove that for every spectral parameter greater than the minimal eigenvalue of the differential part of the equation with the homogeneous Dirichlet condition, the corresponding boundary value problem has a nontrivial strong solution. The corresponding free boundary in this case is of zero measure. A lower estimate for the spectral parameter is established as well. In the latter case the differential part of the equation is formally selfadjoint and the nonlinearity has sublinear growth at infinity. Some upper estimate for the spectral parameter is given in this case.     

Parabolic oblique derivative problem with discontinuous coefficients in generalized Morrey spaces

We study the global Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with $VMO$ coefficients.     

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L^t are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.     

Nodal sets of smooth functions with finite vanishing order and <Emphasis Type="Italic">p</Emphasis>-sweepouts

We show that on a compact Riemannian manifold (M, g), nodal sets of linear combinations of any \(p+1\) smooth functions form an admissible p-sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a result, we obtain a new proof of the Gromov, Guth, Marques–Neves upper bounds on the min–max p-widths of M. We also prove that close to a point at which a smooth function on \(\mathbb {R}^{n+1}\) vanishes to order k, its nodal set is contained in the union of \(k\,W^{1,p}\) graphs for some \(p > 1\). This implies that the nodal set is locally countably n-rectifiable and has locally finite \(\mathcal {H}^n\) measure, facts which also follow from a previous result of Bär. Finally, we prove the continuity of the Hausdorff measure of nodal sets under heat flow.     

Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise and Rough Initial Condition

In this article, we study the parabolic Anderson model driven by a space-time homogeneous Gaussian noise on \(\mathbb {R}_{+} \times \mathbb {R}^d\), whose covariance kernels in space and time are locally integrable nonnegative functions which are nonnegative definite (in the sense of distributions). We assume that the initial condition is given by a signed Borel measure on \(\mathbb {R}^d\), and the spectral measure of the noise satisfies Dalang’s (Electron J Probab 4(6):29, 1999) condition. Under these conditions, we prove that this equation has a unique solution, and we investigate the magnitude of the p-th moments of the solution, for any \(p \ge 2\). In addition, we show that this solution has a Hölder continuous modification with the same regularity and under the same condition as in the case of the white noise in time, regardless of the temporal covariance function of the noise.     

Singular evolution integro-differential equations with kernels defined on bounded intervals

We study linear singular first-order integro-differential Cauchy problems in Banach spaces. The adjective “singular” means here that the integro-differential equation is not in normal form neither can it be reduced to such a form. We generalize some existence and uniqueness theorems proved in [5] Favini, A, Lorenzi, A and Tanabe, H. 2002. Singular integro-differential equations of parabolic type. Advances in Differential Equations, 7: 769–798.  [Google Scholar] for kernels defined on the entire half-line R ₊ to the case of kernels defined on bounded intervals removing the strict assumption that the kernel should be Laplace-transformable.

Particular attention is paid to single out the optimal regularity properties of solutions as well as to point out several explicit applications relative to singular partial integro-differential equations of parabolic and hyperbolic type.     

A Note on Time-Decay Estimates for the Compressible Navier–Stokes Equations

In the recent work, we have developed a decay framework in general L~p critical spaces and established optimal time-decay estimates for barotropic compressible Navier–Stokes equations. Those decay rates of L~q-L~r type of the solution and its derivatives are available in the critical regularity framework, which were exactly firstly observed by Matsumura Nishida, and subsequently generalized by Ponce for solutions with high Sobolev regularity. We would like to mention that our approach is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. In this paper, a new observation is involved in the high frequency, which enables us to improve decay exponents for the high frequencies of solutions.     

L p -regularity for fourth order parabolic systems with measurable coefficients

We treat fourth order parabolic systems in divergence form with bounded measurable coefficients. We establish Hessian estimates for such parabolic systems by proving that the L ^p -regularity of the inhomogeneous terms exactly reflects in the regularity of the Hessian of the solutions for every ${p \in (1, \infty)}$ . The assumptions are that the coefficients are allowed to be merely measurable in one of spatial variables, but averaged in the other spatial variables and time variable.     

$$L^p$$ Christoffel–Minkowski problem: the case $$1< p<k+1$$1<p<k+1

We consider a fully nonlinear partial differential equation associated to the intermediate \(L^p\) Christoffel–Minkowski problem in the case \(1<p<k+1\). We establish the existence of convex body with prescribed k-th even p-area measure on \(\mathbb S^n\), under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain \(C^{1,1}\) regularity estimates for admissible solutions of the equation when \( p\ge \frac{k+1}{2}\).     

<Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript> regularity of solutions of the Levi equation in<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">2n+1</Emphasis></Superscript>

We study the regularity of the solutions of the Levi equation in ?²ⁿ⁺¹. It is a second order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every functionu∈C ². We show that the operator associated to the equation can be represented as a sum of squares of non linear vector fields. Then, by using a freezing method, we prove theC ^∞ regularity of the solutions.     

Maximal regularity of the spatially periodic stokes operator and application to nematic liquid crystal flows

We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal L^p-regularity of the periodic Laplace and Stokes operators and a local-intime existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group \(G: = \mathbb{R}^{n - 1} \times \mathbb{R}/L\mathbb{Z}\) to obtain an R-bound for the resolvent estimate. Then, Weis’ theorem connecting R-boundedness of the resolvent with maximal L^p regularity of a sectorial operator applies.     

Non-uniform hyperbolicity in complex dynamics

We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer n _c so that \(\sum_{n=1}^{\infty} |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles. Let μ _max be the maximal multiplicity of the critical points.The objective is to study the Poincaré series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. If F is summable with exponent \(\alpha<\frac{\delta_{\textit{Poin}}(J)}{\delta_{\textit{Poin}}(J)+\mu_{\textit{max}}}\) where δ _Poin (J) is the Poincaré exponent of the Julia set then there exists a unique, ergodic, and non-atomic conformal measure ν with exponent δ _Poin (J)=HDim(J). If F is polynomially summable with the exponent α, \(\sum_{n=1}^{\infty}n |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles, then F has an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval.We prove that if F is summable with an exponent \(\alpha< \frac{2}{2+\mu_{\textit{max}}}\) then the Minkowski dimension of J is strictly less than 2 if \(J\neq\hat{\mathbb{C}}\) and F is unstable. If F is a polynomial or Blaschke product then J is conformally removable. If F is summable with \(\alpha<\frac{1}{1+\mu_{\textit{max}}}\) then connected components of the boundary of every invariant Fatou component are locally connected. To study continuity of Hausdorff dimension of Julia sets, we introduce the concept of the uniform summability.Finally, we derive a conformal analogue of Jakobson’s (Benedicks–Carleson’s) theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points c from the Mandelbrot set with respect to the harmonic measure.