! !

Advances in Nonlinear PDEs ! in#honor#of##Nina#N.#Uraltseva##

Book$of$abstracts$ St.!Petersburg,!!2014!!!

! ! Advances in Nonlinear PDEs September 3-5, 2014

Advances in Nonlinear PDEs ! Interna2onal#Conference# in#honor#of#the#outstanding#mathema2cian# Nina#N.#Uraltseva## and#on#the#occasion#of#her#80th#anniversary#

St.!Petersburg,!!September!3@5,!2014!!!

Supported!by:!

St.!Petersburg!Steklov!Ins7tute! of!Mathema7cs!

1 Advances in Nonlinear PDEs September 3-5, 2014

! !

Organizing(Commi,ee:(

Darya%Apushkinskaya% Alexander%Mikhailov% Alexander%Nazarov% Sergey%Repin% Ta;ana%Vinogradova% Nadya%Zalesskaya%

2 Advances in Nonlinear PDEs September 3-5, 2014

Cauchy- for quasilinear parabolic systems with a nonsmooth in time principal

Arina Arkhipova St. Petersburg State University, [email protected]

We consider quasilinear parabolic systems with nonsmooth in time principal matrices. Only boundedness of these matrices is assumed in time variable. We prove partial regularity up to the parabolic boundary of a cylinder. To prove the result, we apply the method of modified A-caloric approximation assuming that the matrix A is not the constant one but depends on the time variable. The talk reports on results obtained jointly with J.Stara and O.John.

3 Advances in Nonlinear PDEs September 3-5, 2014

Semilinear elliptic problems with a Hardy potential

Catherine Bandle University of , Switzerland [email protected]

We consider problems of the type p Δu +V (x)u = u in a bounded domain in n where µ V is a Hardy potential and δ(x) is the distance δ 2(x) from a point x to the boundary of the domain. € We are interested in the € existence of positive solutions, and the interplay between the nonlinearity € € and the boundary € singularity. If 0 < p <1 the nonlinearity € gives rise to dead cores and if p >1 to boundary blowup. We give a fairly complete picture of the radial solutions and use those solutions as upper and lower solutions for general domains. € The talk reports on results € obtained in collaboration with V.Moroz (Swansea), W.Reichel (KIT Karslruhe) and M.A.Pozio (La Sapienza Rome)

4 Advances in Nonlinear PDEs September 3-5, 2014

Stochastic counterparts of the Cauchy problem for quasilinear systems of parabolic equations

Yana Belopolskaya St. Petersburg State University for Architecture and Civil Engineering, Russia [email protected]

We consider the Cauchy problem for two types of quasilinear parabolic systems, namely systems with diagonal principal parts and systems with nondiagonal principal parts. For systems of the first type, investigated in the famous monograph [1] we construct probabilistic representations for classical, generalized and viscosity solutions of the Cauchy problem and use them to investigate the PDE system solution. The probabilistic counterpart is constructed in terms of stochastic differential equations for corresponding Markov processes and their multiplicative operator functionals. For systems of the second type studied in a number of papers started from [2] we construct a probabilistic representation of a generalized solution of the Cauchy problem for a PDE system of the second type in terms of time reversed stochastic flows, generated by solutions of corresponding stochastic differential equations. Partial financial support of RFBR Grant 12-01- 00457-a and Minoobrnauki project 1.370.2011 is gratefully acknowledged. References [1] O. Ladyzenskaya, V. Solonnikov, N. Uraltzeva Linear and quasilinear equations of parabolic type 1967, . [2] H. Amann Dynamic theory of quasilinear parabolic systems, Mathematische Zeitschrift (1989), Vol. 202, Issue 2, pp 219- 250.

5 Advances in Nonlinear PDEs September 3-5, 2014

Non-uniqueness of hydrodynamic equations from molecular dynamics

Stamatis Dostoglou University of Missouri-Columbia, USA [email protected]

Hydrodynamic equations can be obtained at the limit of microscopic Newtonian equations as the number of molecules increases while the length scale changes accordingly. For some examples of molecule systems, with reasonable initial conditions and interaction potential, the resulting hydrodynamic equations for certain scaling turn out to be quite easy to analyze and show how microscopic fluctuations can lead to apparent macroscopic non-uniqueness with respect to their macroscopic initial conditions. The contents of the talk are part of an ongoing project to obtain rigorous Reynolds equations and are based on work currently done with Jianfei Xue.

6 Advances in Nonlinear PDEs September 3-5, 2014

Normal equations and nonlocal stabilization by feedback control for equations of Navier- Stokes type

Andrey Fursikov State University, Russia [email protected]

We study so-called parabolic equations of normal type to understand better properties of equations of Navier-Stokes type. By definition semilinear parabolic equation is normal parabolic equation (NPE) if its nonlinear term defined by operator Bsatisfies the condition: ∀v ∈ H1 vector B(v) is collinear to v. In other words solutions of NPE does not satisfies energy estimate “in the most degree". For € Burgers and 3D Helmholtz € equations we € derive normal € parabolic equations (NPE), which nonlinear terms B(v) are orthogonal projections of nonlinear terms for corresponding original equations on the straight line generated by the vector . The structure of dynamical flow corresponding to these NPE v will be described. € For NPE corresponding to Burgers equation we construct nonlocal stabilization to zero of solutions by starting, impulse, or distributed feedback € controls supported in an arbitrary fixed sub domain of the spatial domain. The last result is applied to nonlocal stabilization of solutions for Burgers equations.

7 Advances in Nonlinear PDEs September 3-5, 2014

On attractors of m-Hessian evolutions

Nina Ivochkina St. Petersburg State University, Russia [email protected]

Let Ω be a bounded domain in n, 2,1 Q = Ω × (0;∞), u ∈C (Q ), uxx be the Hesse matrix

of u in space variables. We denote by Tp[u] = Tp (uxx ), 1 p n $1 the p-trace of u and introduce p- ≤€ ≤ xx € Hessian evolution operator by € E [u]€:= u T [u]+€T [u]. € p t p −1 p We investigate € asymptotic behavior of € solutions € of the following € initial boundary € value problems: E [u] = f , u = φ, 1≤ m ≤ n, (1) € m ∂ 'QT where . In ∂'QT = {Ω × {0}} ∪{∂Ω × [0;T]} particular, we have proved € 2,1 Theorem 1. Let f ≥ν > 0, f ∈C (QT ) for all € 2,1 T ∈[0;∞), φ ∈C (QT ), φ = 0 on ∂Ω × [0;∞), 2 ∂Ω ∈C . Assume that limt → ∞ f (x,t) = f (x) and 2 there exists € a solution € u ∈C (Ω ) to the Dirichlet problem € € € € T [u] = f , u = 0. € € m ∂Ω Then all solutions u ∈C 2,1(Ω × [0;∞)) to the problem (1) tend uniformly € in C to the function u (x) , when t →∞ . € It is of interest the following non existence theorem. € € € €

8 Advances in Nonlinear PDEs September 3-5, 2014

Theorem 2. Assume that there are points x0, x1 ∈Ω

such that φxx (x0,0) is (m −1)-positive matrix, while

φxx (x1,0) is not (m −1)-positive. Then there are no solutions in C 2,1(Q ) to the problem (1), whatever T € f > 0, ∂Ω, T > 0, φ had been. € € € €Eventually, we formulate the existence theorem assuming sufficiently smooth€ data in (1). € € € € Theorem 3. Let f ≥ν > 0, ∂Ω is ( m −1)-convex

hypersurface, φ(x,0) ∈K m −1(Ω ). Assume that compatibility conditions are satisfied. Then there exists a unique in C 2,1(Q ) solution to the problem (1). T € € € € The work is supported by the RFBR grant No. 12-01-00439 and by the grants Sci. Schools RF 1771.2014.1,€ and by the St. Petersburg State University grant 6.38.670.2013.

9 Advances in Nonlinear PDEs September 3-5, 2014

Analyticity of the free boundary in the thin obstacle problem

Herbert Koch University of Bonn, Germany [email protected]

We prove analyticity of the regular part of the boundary of the contact set for the thin obstacle problem. A key step is a change of variables which reduces the problem to a Monge-Ampere type equation, which in the relevant regime becomes a fully nonlinear variant of a Grushin-type operator. We prove analyticity of solutions to this fully nonlinear equation under non degeneracy assumptions. This talk is based on results obtained in collaboration with A.Petrosyan and W.Shi.

10 Advances in Nonlinear PDEs September 3-5, 2014

On uniqueness in the water wave theory

Vladimir Kozlov Linköping University, [email protected]

This talk concerns the classical free-boundary problem that describes two-dimensional steady gravity waves on water of finite depth. I will discuss some uniqueness results.

11 Advances in Nonlinear PDEs September 3-5, 2014

Common features of homogenization and of large scale limits in statistical mechanics

Stephan Luckhaus Leipzig University, Germany [email protected]

We try to present large scale limits in equilibrium statistical mechanics from an analytic- functional analytic point of view. The results presented are joint with Roman Kotecky. We also try to point out the analogues with results in homogenization. The setting for both types of result is that of quasiconvex functionals.

12 Advances in Nonlinear PDEs September 3-5, 2014

Criteria for the Poincare-Hardy inequalities

Vladimir Maz’ya , UK and Linköping University, Sweden [email protected]

This is a survey of necessary and sufficient conditions for validity of various integral inequalities containing arbitrary weights (measures and distributions). These results have direct applications to the spectral theory of elliptic partial differential operators.

13 Advances in Nonlinear PDEs September 3-5, 2014

From Uraltseva to Zhikov, forty years of degenerate operators in Russia

Giuseppe Mingione University of Parma, Italy [email protected]

In 1967 a seminal paper of Nina Ural'tseva appeared, featuring the proof of the C1,β -nature of solutions to the degenerate equation p-Laplacian equations. This is a cornerstone of modern nonlinear and the techniques introduced by Nina are nowadays classical. € Several years later, € Zhikov and a group of Russian developed new models for strongly anisotropy materials, based on the p- Laplacian operator, in order to study various questions in homogenisation, elasticity, Lavrentiev phenomenon. I will present a few new regularity results on such functionals. €

14 Advances in Nonlinear PDEs September 3-5, 2014

The Dirichlet and Navier fractional Laplacians

Roberta Musina Udine University, Italy [email protected]

This talk is based on results obtained jointly with A.I. Nazarov. Let Ω ⊂n be a bounded and smooth domain.

We denote by λ j and ϕ j the eigenvalues and 1 eigenfunctions of the Laplace operator −Δ on H0(Ω) , respectively. € For any real number s > 0 we formally introduce the "Dirichlet" fractional Laplacian € € s F (−Δ) u€(ξ ) =| ξ |€2s F[u](ξ), [ D ] where F is the Fourier transform, and the "Navier" € fractional Laplacian ⎛ ⎞ € (−Δ )s u = λs uϕ ϕ . Ω N ∑ j ⎜ j ∫ j ⎟ j ⎝ Ω ⎠ € In [MN1] and [MN2] we compare those two fractional Laplacians for arbitrary s > 0. In this talk we focus our attention to the case s ∈(0,1), when the domains of the € s s quadratic forms (−Δ)D u,u and (−Δ Ω )N u,u coincide with € ˜ s s n H€ (Ω) = {u ∈ H ( ) | supp(u) ⊆ Ω} . The following facts are proved: € € s s a. The operator (−Δ Ω )N − (−Δ)D is positive definite and positivity preserving. € € s b. For any fixed u ∈ H˜ (Ω) one has s s (−Δ)D u,u = inf (−Δ Ω )N u,u € Ω'⊃Ω

€ € 15 Advances in Nonlinear PDEs September 3-5, 2014

(the infimum is taken over the family of smooth bounded domains). c. Assume n ≥ 2 or s <1/2, and put 2n 2* = . Then the "Dirichlet-Sobolev" and s n − 2s the "Navier-Sobolev" constants coincide, that is, € € ( )s u,u ( )s u,u −Δ D −Δ Ω N inf 2 = inf 2 . u H˜ s ( ), u 0 u H˜ s ( ), u 0 € ∈ Ω ≠ u * ∈ Ω ≠ u * 2s 2s d. The "Dirichlet-Hardy" and the "Navier-Hardy" constants coincide as well:

€ s s (−Δ)D u,u (−Δ Ω )N u,u inf 2 = inf 2 . u∈H˜ s (Ω), u≠0 | x | −s u u∈H˜ s (Ω), u≠0 | x | −s u 2 2

References:

[MN1] € R. Musina, A.I. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eqs, vol. 39 (2014), no. 9, 1780--1790. [MN2] R. Musina, A.I. Nazarov, On fractional Laplacians-II, preprint (2014).

16 Advances in Nonlinear PDEs September 3-5, 2014

On the Cauchy problem for scalar balance laws in the class of Besicovitch almost periodic functions

Evgeny Panov Novgorod State University, Russia [email protected]

We study the Cauchy problem for a first order quasilinear equation (a balance law) with a merely continuous flux vector and almost periodic (in Besicovitch sense) initial and source functions. It is proved that the Kruzhkov entropy solution is almost periodic with respect to the spacial variables, it is unique (as a function with values in the Besicovitch space), and its spectrum is contained in the additive subgroup generated by the spectra of initial and source data. A precise nondegeneracy condition is also found for large-time decay property of entropy solutions.

17 Advances in Nonlinear PDEs September 3-5, 2014

Homogenization of monotone operators with oscilating exponent of nonlinearity

Svetlana Pastukhova Moscow Institute of Radioengineering, Electronic and Automation, Russia [email protected]

We consider the Dirichlet problem for an elliptic monotone-type equation under coerciveness

and growth conditions with a variable exponent pε (x). Here p (x) = p x and p(y) is a measurable ε ( ε ) periodic function such that 1 < α ≤ p(y) ≤ β < ∞. When the small parameter ε > 0 tends to zero, the € exponent pε (x)$ is highly oscillating and we need to € homogenize the problem passing to the limit as € ε →0. Generally, we have € here the Lavrent’ev phenomenon and solutions of two € different types should be taken €into consideration. These are so-called W - and H - solutions considered, respectively, in € the most broad and in the most narrow Sobolev spaces connected with

the exponent pε (x). For each type of solutions, we give the homogenization procedure € and € describe the limit problem. This is a joint result with prof. V.V. Zhikov. €

18 Advances in Nonlinear PDEs September 3-5, 2014

Viscous incompressible free-surface flow down an inclined perturbed plane

Konstantinas Pileckas Vilnius University, Lithuania [email protected]

The stationary plane free boundary value problems for the Navier-Stokes equations is studied. The problem models the viscous fluid free- surface flow down a perturbed inclined plane. For sufficiently small data the solvability and uniqueness results are proved in Hölder spaces. The asymptotic behaviour of the solution is investigated. This is a joint work with V.A. Solonnikov.

19 Advances in Nonlinear PDEs September 3-5, 2014

On the regularity and stability of the free boundary of obstacle type heterogeneous problems

José Francisco Rodrigues CMAF, University of Lisbon, Portugal [email protected]

We extend some properties to the solutions of free boundary problems of obstacle type with two phases for a class of heterogeneous quasilinear elliptic operators. Under a natural non degeneracy assumption on the interface, corresponding to the zero level set, we prove a continuous dependence result for the characteristic functions of each phase and we establish sharp estimates on the variation of its Lebesgue measure with respect to the L1-variation of the data, in a rather general framework. For elliptic quasilinear equations that have solutions with integrable second order derivatives, we show that the characteristic € functions of both phases are of bounded variations for non degenerating heterogeneous forces. This extends recent results for the obstacle problem and is a first result on the regularity of the free boundary of the heterogeneous two phases problem.

20 Advances in Nonlinear PDEs September 3-5, 2014

Leray-Hopf solutions to Navier-Stokes equations with weakly converging initial data

Gregory Seregin University of Oxford, UK and St. Petersburg Steklov Institute of , Russia [email protected]

The talk is addressed the question about convergence of a sequence of week Leray-Hopf solutions to the initial for the 3D Navier-Stokes equations provided that the corresponding initial data converge weakly to their limit. Under certain rather mild assumptions, it is shown that the limit velocity field is a weak Leray-Hopf solution with the limit initial data.

21 Advances in Nonlinear PDEs September 3-5, 2014

Logarithmic interpolation-embedding inequalities on irregular domains

Tatyana Shaposhnikova Royal Institute of Technology, Sweden [email protected]

Logarithmic interpolation-embedding inequalities of Brezis-Gallouet-Wainger type are proved for various classes of irregular domains, in particular, for power cusps and lambda-John domains. This is a joint work with Vladimir Maz'ya.

22 Advances in Nonlinear PDEs September 3-5, 2014

Lp -estimates of solutions of some problems of hydrodynamics and magnetohydrodynamics

€ Vsevolod Solonnikov St. Petersburg Steklov Institute of Mathematics, Russia [email protected]

We discuss Lp -estimates of solutions of some initial-boundary value problems arising in the analysis of motion of an isolated liquid mass of viscous incompressible electrically conducting fluid. €

23 Advances in Nonlinear PDEs September 3-5, 2014

Regularity of solutions to elliptic and parabolic systems with L1 right hand side

Jana Stará Charles University, Czech Republic€ [email protected]

The talk is devoted to extension of the results of P. Baroni, J. Habermann for one nonlinear parabolic equation (published in 2012) to parabolic systems. We wanted to avoid stronger ellipticity assumption and we used an approach based on ideas of G. Stampacchia. This approach uses duality to define a type of very weak solution. As a consequence we restricted ourselves to the study of existence and regularity properties of solutions to linear parabolic systems with non smooth coefficients and right hand sides. We consider the parabolic system ∂u − div(ADu) = f in Q, ∂t u = 0 on ∂pQ, where Q = (0,T) × Ω, Ω is a C1 bounded domain in n  and ∂pQ is the parabolic boundary of Q. We obtain existence of very weak solutions for coefficient matrix A that satisfies standard ellipticity and boundedness € € conditions and € whose € entries belong to Sarason space € VMO (Q). € We prove that for any right hand side € f ∈ L1(Q) there exists unique very weak solution u r 1,q n such that u ∈ L (0,T;W0 (Ω;  )) for any r, q ≥1; € 2 n r + q ≥ n +1. € € € € € € 24 Advances in Nonlinear PDEs September 3-5, 2014

For "better" right hand side f in Morrey space 1,τ q,ν n L (Q) we prove that Du ∈ Lloc (Q);  with n +2 q ∈(1, n +1 ),ν = n + 2 − q[n +1−τ(n + 2)]. Moreover, for coefficients € that are α -Hölder continuous in space variables we prove fractional € differentiability € of space gradient of u, i.e. η, η / 2,q,τ € Du€ ∈Wloc with a positive number η which depends on the parameters of the problem. €

€

€ €

25 Advances in Nonlinear PDEs September 3-5, 2014

Properties of the free boundary in the optimal compliance problem

Eugene Stepanov St. Petersburg Steklov Institute of Mathematics, Russia [email protected]

We consider the problem of finding the optimal shape of a support of some elastic material under the given load so as to optimize the compliance of the latter. The topological properties of the optimal support (which may be viewed as the “free boundary”) as well as its regularity will be studied.

26 Advances in Nonlinear PDEs September 3-5, 2014

A chemotaxis-model with non-diffusing attractor

Angela Stevens University of Münster, Germany [email protected]

Cell motion due to attractive chemicals is a widespread phenomenon in developmental biology. A classical example is chemotaxis, which - among others - is modeled via the well known Keller-Segel cross- diffusion system. In this talk a variant of this mathematical model is presented, namely a system where the attractive chemical is not diffusing. So an ODE instead of a reaction-diffusion equation is coupled to the nonlinear chemotaxis equation for the cells. This model is analyzed w.r.t. the existence of global solutions and blow-up. The PDE-ODE system behaves very different in this respect if compared with the classical Keller-Segel model for chemotaxis.

27 Advances in Nonlinear PDEs September 3-5, 2014

Homogenization of periodic elliptic operators: error estimates in dependence of the spectral parameter

Tatiana Suslina St. Petersburg State University, Russia [email protected]

We study matrix elliptic second order d differential operators Aε , ε > 0, in  or in a bounded domain O ⊂d (with sufficiently smooth boundary) with the Dirichlet or Neumann boundary conditions. It * is assumed that Aε = b(D) g(x /ε)b(D), where a (m × m)-matrix€ €- valued function g(x) is bounded, € uniformly positive definite and periodic with respect to d some lattice; b(D) = b D is (m × n)-matrix € ∑ j =1 j j € first order . € It is assumed that d m ≥ n and the symbol b(ξ) = b ξ has maximal ∑ j =1 j j rank. € € We study the behavior of the resolvent −1 for small . It turns out that this resolvent € (Aε −ζ I) ε € 0 −1 converges to (A −ζ I) in the L2-operator norm, as ε →0, where A0 = b(D)* g0b(D) is the effective operator. We find twoparametric error estimates for € € the norm of the difference (A −ζ I)−1 − (A0 −ζ I)−1 € ε with respect to € ε and ζ. Also, we find approximation € € −1 1 for the resolvent (Aε −ζ I) in the (L2 →H )- operator norm with twoparametric error estimates. In this approximation, € the corrector term is taken into account. € The results can be applied to homogenization € of parabolic initial boundary€ -value problems.€

28 Advances in Nonlinear PDEs September 3-5, 2014

Free boundary problems for mechanical models of tumor growth

Juan Luis Vazquez Autonomous University of Madrid, Spain [email protected]

Mathematical models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. We first formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain singular limit which leads to a Hele-Shaw type problem. A detailed of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the limit problem. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. Using this theory as a basis, we go on to consider a more complex model including nutrients. Here, technical difficulties appear, that reduce the generality and detail of the description. We prove uniqueness for the system, a main mathematical difficulty. Joint work with Benoit Perthame, from , and Fernando Quiros, from Madrid.

29 Advances in Nonlinear PDEs September 3-5, 2014

On density of smooth functions in weighted Sobolev Spaces

Vasily Zhikov Vladimir State University, Russia [email protected]

30