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We describe regularizing effects in the linearization of a kinetic equation for nonlinear waves satisfying the Schrödinger equation in terms of weak turbulence and condensate. The problem is first considered in spaces of bounded functions with weights, where existence of solutions and some first regularity properties are proved. After a suitable change of variables the equation is written in terms of a pseudo differential operator. Homogeneity of the equation and classical arguments of freezing of coefficients may then be used to prove regularizing effect in local Sobolev type spaces. [ABSTRACT FROM AUTHOR]

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω−⊂ℝ3$$ {\Omega}^{-}\subset {\mathrm{\mathbb{R}}}^3 $$, has a compact boundary surface S=∂Ω−$$ S=\partial {\Omega}^{-} $$, which is divided into two disjoint parts, the Dirichlet part SD$$ {S}_D $$ and the Neumann part SN$$ {S}_N $$, where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L2$$ {L}_2 $$‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space [W2,loc1(Ω−)]3$$ {\left[{W}_{2, loc}^1\left({\Omega}^{-}\right)\right]}^3 $$ and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the Lp$$ {L}_p $$‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses Cα$$ {C}^{\alpha } $$‐Hölder continuity property in the closed domain Ω−‾$$ \overline{\Omega^{-}} $$ with α=12−ε$$ \alpha =\frac{1}{2}-\varepsilon $$, where ε>0$$ \varepsilon >0 $$ is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

In this paper we discuss a class of spectral partition problems with a measure constraint, for partitions of a given bounded connected open set. We establish the existence of an optimal open partition, showing that the corresponding eigenfunctions are locally Lipschitz continuous, and obtain some qualitative properties for the partition. The proof uses an equivalent weak formulation that involves a minimization problem of a penalized functional where the variables are functions rather than domains, suitable deformations, blowup techniques, and a monotonicity formula. [ABSTRACT FROM AUTHOR]

We consider the nonlinear elliptic system $$\begin{align*} \left\{ \begin{array}{ll} u \in W_{0}^{\frac{N}{N-1}}(\Omega): & -\textrm{div}(M(x)\nabla u) + u = -\textrm{div}(u\,M(x)\,\nabla\psi) + f(x),\\ \displaystyle \psi \in W_0^{1,2}(\Omega): & -\textrm{div}(M(x)\nabla \psi) + \psi = R(u) + E(x)\,\nabla\psi, \end{array} \right. \end{align*}$$ { u ∈ W 0 N N − 1 (Ω) : − div (M (x) ∇u) + u = − div (u M (x) ∇ψ) + f (x) , ψ ∈ W 0 1 , 2 (Ω) : − div (M (x) ∇ψ) + ψ = R (u) + E (x) ∇ψ , where Ω is a bounded, open subset of $ \mathbb {R}^N $ R N , $ N \geq ~3 $ N ≥ 3 ; $ M(x) $ M (x) is a coercive, symmetric matrix with $ L^{\infty }(\Omega) $ L ∞ (Ω) coefficients; $ f(x) $ f (x) and $ E(x) $ E (x) belong to some Lebesgue space, and $ R(s) $ R (s) is a continuous function such that $$\begin{align*} 0 \leq R(s)\leq |s|^{\theta}, \quad \mbox{for}\ \theta { \lt } \frac{2}{N}. \end{align*}$$ 0 ≤ R (s) ≤ | s | θ , for θ < 2 N. Using a duality technique, we prove existence of at least a weak solution $ (u,\psi) $ (u , ψ). Moreover, if N=3 or N=4, we prove under stronger assumptions on $ f(x) $ f (x) and $ E(x) $ E (x) that the solution u belongs to $ W_0^{1,2}(\Omega) $ W 0 1 , 2 (Ω). [ABSTRACT FROM AUTHOR]

The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both the state and adjoint state equations. The article establishes the existence and uniqueness of solutions for both equations, with minimal assumptions made about the problem's data. Next, the regularity of these solutions is studied in three frameworks: Hilbert–Sobolev spaces, Sobolev–Slobodeckiĭ spaces, and weighted Sobolev spaces. These regularity results enable a numerical analysis of the finite element approximation of both the state and adjoint state equations. The results cover both convex and non-convex domains and quasi-uniform and graded meshes. Finally, the optimal control problem is analyzed and discretized. Existence and uniqueness of the solution, first-order optimality conditions, and error estimates for the finite element approximation of the control are obtained. Numerical experiments confirming these results are included. [ABSTRACT FROM AUTHOR]

The paper studies three-dimensional interaction crack type boundary-transmission problems of pseudo-oscillations between thermo-elastic and thermo-piezo-elastic bodies taking microrotations into account. The model under consideration is based on the Green–Naghdi theory of thermo-piezo-electricity without energy dissipation. This theory permits the thermal waves to propagate only with a finite speed. The system of partial differential equations of pseudo-oscillations is obtained from the corresponding dynamical model by the Laplace transform. Using the potential theory and the method of boundary pseudodifferential equations, we prove the existence, uniqueness and regularity of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the influence of the initial data and the forcing terms on the regularity of solutions to a class of evolution equations including linear and semilinear parabolic equations as the model cases, together with the nonlinear p-Laplacian equation. We focus our study on the regularity (in terms of belonging to appropriate Lebesgue spaces) of the gradient of the solutions. We prove that there are cases where the regularity of the solutions as soon as  t > 0  is not influenced at all by the initial data. We also derive estimates for the gradient of these solutions that are independent of the initial data and reveal, once again, that for this class of evolution problems, the real "actors of the regularity" are the forcing terms. [ABSTRACT FROM AUTHOR]

In this paper, we address an optimal distributed control problem for a non-local model of phase-field type, describing the evolution of tumour cells in presence of a nutrient. The model couples a non-local and viscous Cahn–Hilliard equation for the phase parameter with a reaction-diffusion equation for the nutrient. The optimal control problem aims at finding a therapy, encoded as a source term in the system, both in the form of radiotherapy and chemotherapy, which could lead to the evolution of the phase variable towards a desired final target. First, we prove strong well-posedness for the system of non-linear partial differential equations. In particular, due to the presence of a viscous regularisation, we can also consider double-well potentials of singular type and cross-diffusion terms related to the effects of chemotaxis. Moreover, the particular structure of the reaction terms allows us to prove new regularity results for this kind of system. Then, turning to the optimal control problem, we prove the existence of an optimal therapy and, by studying Fréchet-differentiability properties of the control-to-state operator and the corresponding adjoint system, we obtain the first-order necessary optimality conditions. [ABSTRACT FROM AUTHOR]

study minimizers of the energy functional ... where F (u) ≈ |u|q log u for some -1 < q < 0. We prove existence, optimal decay, and non-degeneracy of solutions, from free boundary points. Consequently, we derive the porosity property and an estimate on the Hausdorff dimension of the free boundary. [ABSTRACT FROM AUTHOR]

For the case 0 <\α<1/3, we construct unique solutions to the fractal Burgers' dtu + udxu + (-\γ)αu = 0 which develop a first shock in finite time, starting from smooth generic initial data. This first singularity is an asymptotically self-similar, stable H6 perturbation of a stable, self-similar Burgers' shock profile. Furthermore, we are able to compute the spatio-temporal location and H\6older regularity for the first singularity. There are many results showing that gradient blowup occurs in finite time for the supercritical range, but the present result is the first example where singular solutions have been explicitly constructed and thus precisely characterized. [ABSTRACT FROM AUTHOR]

In this paper we establish some regularity results concerning the behavior of weak solutions and very weak solutions of the degenerate wave equation near the boundary. For the nondegenerate case, the correponding results were originally obtained by Fabre and Puel (J. of Diff. Eq. 106, 1993). This kind of results is closely related to the exact boundary controllability for the wave equation as the limit of internal controllability. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with global maximal regularity estimates for elliptic equations with degenerate weights. We consider both the linear case and the non-linear case. We show that higher integrability of the gradients can be obtained by imposing a local small oscillation condition on the weight and a local small Lipschitz condition on the boundary of the domain. Our results are new in the linear and non-linear case. We show by example that the relation between the exponent of higher integrability and the smallness parameters is sharp even in the linear or the unweighted case. [ABSTRACT FROM AUTHOR]

We consider a special approach to investigate a mixed boundary value problem (BVP) for the Lamé system of elasticity in the case of three-dimensional bounded domain Ω ⊂ R 3 , when the boundary surface S = ∂ Ω is divided into two disjoint parts, S D and S N , where the Dirichlet and Neumann type boundary conditions are prescribed respectively for the displacement vector and stress vector. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations which do not contain neither extensions of the Dirichlet or Neumann data, nor the Steklov–Poincaré type operator. Moreover, the right hand sides of the resulting pseudodifferential system are vectors coinciding with the Dirichlet and Neumann data of the problem under consideration. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space [ W 2 1 (Ω) ] 3 and representability of solutions in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that the operator is invertible in the L p -based Besov spaces with 4 3 < p < 4 , which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ‾ with α = 1 2 − ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

We show that the bounded solutions to the fractional Helmholtz equation, (− Δ) s u = u for 0 < s < 1 in ℝ n , are given by the bounded solutions to the classical Helmholtz equation (− Δ) u = u in ℝ n for n ≥ 2 when u is additionally assumed to be vanishing at ∞. When n = 1 , we show that the bounded fractional Helmholtz solutions are again given by the classical solutions A cos x + B sin x. We show that this classification of fractional Helmholtz solutions extends for 1 < s ≤ 2 and s ℕ when u C ∞ (ℝ n). Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation ψ (− Δ) u = ψ (1) u in ℝ n , when ψ is complete Bernstein and certain regularity conditions are imposed on the associated weight a (t). [ABSTRACT FROM AUTHOR]

In this note, we describe a way to study local regularity of a weak solution to the Navier-Stokes equations, satisfying the simplest scale-invariant restriction, with the help of zooming and duality approach to the corresponding mild bounded ancient solution. [ABSTRACT FROM AUTHOR]

We investigate the existence of multiple solutions for the (p, q)- quasilinear elliptic problem {−∆pu − ∆qu = g(x, u) + ε h(x, u) in Ω, u = 0 on ∂Ω, where 1 < p < q < +∞, Ω is an open bounded domain of RN, the nonlinearity g(x, u) behaves at infinity as |u| q−1, ε ∈ R and h ∈ C(Ω × R, R). In spite of the possible lack of a variational structure of this problem, from suitable assumptions on g(x, u) and appropriate procedures and estimates, the existence of multiple solutions can be proved for small perturbations. [ABSTRACT FROM AUTHOR]

In this paper, the Harnack inequality and Hölder continuity results are established for a new class of degenerate parabolic equations ∂ x i (a i j (t , x) ∂ x j u) − ∂ t u = 0 on a bounded domain D ⊂ R n + 1. The coefficients matrix A = ‖ a i j (t , x) ‖ , i , j = 1 , 2 ,... n is real, symmetric, positive and such that there exist positive constants c 1 , c 2 with c 1 ω (t , x) | ξ | 2 ≤ a i j (t , x) ξ i ξ j ≤ c 2 ω (t , x) | ξ | 2 for ∀ ξ ∈ R n and arbitrary (t , x) ∈ D. The exclusive Muckenhoupt condition ω p ∈ A 1 + p / r : (⨍ Q T ω p d x d t) 1 / p (⨍ Q T σ r d x d t) 1 / r ≤ c 0 is assumed for a p > (n + 2) / 2 , an r > n / 2 such that n / 2 r + (n + 2) / 2 p < 1 ; the cylinders { Q T = K R x 0 × (t 0 − T , t 0) : (t 0 , x 0) ∈ D , T = C R 2 / (⨍ Q T ω p d x d t) 1 / p } , σ = 1 / ω , K r x = { y ∈ R n : | y − x | < r }. The cylinders are of T → 0 as R → 0. [ABSTRACT FROM AUTHOR]

We derive a fundamental solution E to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of E as well as formulas for solutions to the Dirichlet and fixed slope problems in terms of convolution of E with data. We also study integrability of derivatives of solutions given in this way. We present conditions, which are sufficient for uniqueness of solutions. Finally, we show the infinite speed of signal propagation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 , when the boundary surface S = ∂ ⁡ Ω is divided into two disjoint parts S D and S N where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ⁢ (Ω) . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p -based Besov spaces, which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ with α = 1 2 - ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

We consider the problem of existence of a (unique) weak solution to the SDE describing symmetric α‐stable process with a locally unbounded drift b:Rd→Rd$b:\mathbb {R}^d \rightarrow \mathbb {R}^d$, d≥3$d \ge 3$, 1<α<2$1<\alpha <2$. In this paper, b belongs to the class of weakly form‐bounded vector fields, the class providing the L2 theory of the non‐local operator behind the SDE, i.e. (−Δ)α2+b·∇$(-\Delta)^{\frac{\alpha }{2}} + b \cdot \nabla$. It contains as proper sub‐classes other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, the weak Ld/(α−1)$L^{\scriptscriptstyle d/(\alpha -1)}$ class and the Campanato–Morrey class (in general, such b makes invalid the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that the operator (−Δ)α2+b·∇$(-\Delta)^{\frac{\alpha }{2}} + b \cdot \nabla$ with weakly form‐bounded b admits a realization as (minus) Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of (−Δ)α2+b·∇$(-\Delta)^{\frac{\alpha }{2}} + b \cdot \nabla$ in Lp$L^p$, p>d−α+1$p>d-\alpha +1$. [ABSTRACT FROM AUTHOR]

In Gromov and Schoen’s (Publications Mathématiques de l’Institut des Hautes Études Scientifiques 76(1):165–246, 1992), a notion of energy-minimizing maps into singular spaces is developed. Using some analogs of classical tools, such as Almgren’s frequency function, they focus on harmonic maps into F-connected complexes, and demonstrate that the singular sets of such functions are closed and of codimension 2. More recent work on related problems, such as Naber and Valtorta’s (Ann Math 185(1):131–227, 2017), uses the study of monotone quantities such as the frequency function to prove stronger rectifiability results about the singular sets in these contexts. In this paper, we adapt this more recent work to demonstrate that an energy-minimizing map from a Euclidean space R m to an F-connected complex has a singular set which is (m - 2) -rectifiable. We also obtain Minkowski bounds on the singular sets of these maps, and in particular, use these bounds to demonstrate the local finiteness of the (m - 2) -dimensional Hausdorff measure of these singular sets. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special approach to investigate a three-dimensional mixed boundary value problem (BVP) for an anisotropic Helmholtz type equation which is a second order elliptic partial differential equation containing a complex parameter ν. The boundary surface S = @ of a domain under consideration, = R3, is divided into two disjoint parts, SD and SN, where the Dirichlet and Neumann type boundary conditions are prescribed respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of linear combination of the corresponding single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP to a system of pseudodifferential equations. It is shown that the corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L2-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W1 2 (). Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the Lp-based Besov spaces, which under appropriate boundary data implies C-Höolder continuity of the solution to the mixed BVP in the closed domain with α = 1/2 -, where" > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

In R d , d ≥ 3 , consider the divergence and the non-divergence form operators - Δ - ∇ · (a - I) · ∇ + b · ∇ , - Δ - (a - I) · ∇ 2 + b · ∇ , where the second-order perturbations are given by the matrix a - I = c | x | - 2 x ⊗ x , c > - 1. The vector field b : R d → R d is form-bounded with form-bound δ > 0 . (This includes vector fields with entries in L d , as well as vector fields having critical-order singularities.) We characterize quantitative dependence on c and δ of the L q → W 1 , q d / (d - 2) regularity of solutions of the corresponding elliptic and parabolic equations in L q , q ≥ 2 ∨ (d - 2) . [ABSTRACT FROM AUTHOR]

In this paper we obtain the local Hölder regularity of the weak solutions for singular parabolic equations with variable exponents. The proof is based on DiBenedetto’s technique called intrinsic scaling; by choosing an appropriate geometry one can deduce energy and logarithmic estimates from which one can implement an iterative method to obtain the regularity result.

The purpose of this note is to present a “new” approach to the decay rate of the solutions to the no-sign obstacle problem from the free boundary, based on Weiss-monotonicity formula. In presenting the approach we have chosen to treat a problem which is not touched earlier in the existing literature. Although earlier techniques may still work for this problem, we believe this approach gives a shorter proof, and may have wider applications.

This manuscript is dedicated to Umberto Mosco, with esteem and affection. Umberto was my mentor at the University of Rome, where I completed my four years studies in Mathematics before my PhD program in Pisa. I dedicate to him the article, which is divided in two parts. In the first section I propose some regularity theorems, precisely some interior bounds for the gradient of weak solutions to a class of nonlinear elliptic systems; the title of this manuscript takes its origin from this section. The second part of the manuscript deals with my first studies in Rome together with Umberto Mosco and with my next studies in Pisa where I met Ennio De Giorgi and where I had the good fortune of assisting to the birth of the G−convergence and the Γ−convergence theories, with some connections with the Mosco’s convergence.

Slightly compressible Brinkman–Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the existence of a global and an exponential attractors for these equations in a natural phase space is verified. [ABSTRACT FROM AUTHOR]

We extend to the singular case the results in [El Bahja H. Regularity for anisotropic quasi-linear parabolic equations with variable growth. Electron J Differ Equ. 2019;2019(104):1–21] concerning the regularity of weak solutions of a class of anisotropic quasi-linear parabolic equations with variable exponents. The proof is based on DiBenedetto's technique called intrinsic scaling; by choosing an appropriate geometry one can deduce energy and logarithmic estimates from which one can implement an iterative method to obtain the regularity result. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems. [ABSTRACT FROM AUTHOR]

We consider the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility in a bounded domain Ω ⊂ R d , d ≤ 3. We first prove the existence of maximal strong solutions in weighted (in time) L p spaces. Then we establish further regularity properties of the solution through maximal regularity theory. Finally, we revisit the separation property in an appendix. [ABSTRACT FROM AUTHOR]

In this note we are concerned with interior regularity properties of the p-Poisson problem Δp(u) = f with p > 2. For all 0 < λ ≤ 1 we constuct right-hand sides f of differentiability −1 + λ such that the (Besov-) smoothness of corresponding solutions u is essentially limited to 1 + λ/(p − 1). The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savaré [J. Funct. Anal. 152 (1998), 176–201], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130 (2016), 298–329]. [ABSTRACT FROM AUTHOR]

In this paper we study the regularity and the behavior in time of the solutions to a quasilinear class of noncoercive problems whose prototype is u t - div (a (x , t , u) ∇ u) = - div (u E (x , t)) in Ω × (0 , T) , u (x , t) = 0 on ∂ Ω × (0 , T) , u (x , 0) = u 0 (x) in Ω. In particular we show that under suitable conditions on the vector field E, even if the problem is noncoercive and although the initial datum u 0 is only an L 1 (Ω) function, there exist solutions that immediately improve their regularity and belong to every Lebesgue space. We also prove that solutions may become immediately bounded. Finally, we study the behavior in time of such regular solutions and we prove estimates that allow to describe their blow-up for t near zero. [ABSTRACT FROM AUTHOR]

The paper considers a class of linear Boltzmann transport equations which models charged particle transport, for example in dose calculation of radiation therapy. The equation is an approximation of the exact transport equation containing hyper-singular integrals in its collision terms. The paper confines to the global case where the spatial domain G is the whole space R 3. Existence results of solutions for the due initial value problem are formulated by applying variational methods. In addition, some regularity results of solutions are verified in scales of relevant anisotropic mixed-norm Sobolev spaces. [ABSTRACT FROM AUTHOR]

We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators. More specifically, we consider viscosity solutions of ... with γ>0, ϕ∈C1,α(B1) for some α∈(0,1], a continuous boundary datum g and f∈L∞(B1)∩C0(B1) and prove that they are C1,β(B1/2) (and in particular at free boundary points) where β=min{α,1γ+1}. Moreover, we achieve such a feature by using a recently developed geometric approach which is a novelty for these types of free boundary problems. Furthermore, under a natural non-degeneracy assumption on the obstacle, we prove that the free boundary ∂{u>ϕ} has Hausdorff dimension less than n (and in particular zero Lebesgue measure). Our results are new even for degenerate problems such as |Du|γu=χ{u>ϕ} with γ>0. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish new regularity results for a non-isothermal Cahn-Hilliard system in the two dimensional setting. The main achievement is a crucial estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn–Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Meth. Biomed. Eng.28 (2012), 3–24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn–Hilliard equation for the tumor cell fraction φ, coupled to a reaction–diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases. [ABSTRACT FROM AUTHOR]

We study a parabolic equation with the data and the coefficient of zero order term only summable functions. Despite all this lack of regularity we prove that there exists a solution which becomes immediately bounded. Moreover, we study the asymptotic behavior of this solution in the autonomous case showing that the constructed solution tends to the associate stationary solution. [ABSTRACT FROM AUTHOR]

In this paper we obtain the local Hölder regularity of the weak solutions for singular parabolic equations with variable exponents. The proof is based on DiBenedetto's technique called intrinsic scaling; by choosing an appropriate geometry one can deduce energy and logarithmic estimates from which one can implement an iterative method to obtain the regularity result. [ABSTRACT FROM AUTHOR]

We deal with the following Cauchy problem for a Schrödinger equation: D t u - Δ u + ∑ j = 1 n a j (t , x) D x j u + b (t , x) u = 0 , u (0 , x) = g (x). We assume a decay condition of type | x | - σ , σ ∈ (0 , 1) , on the imaginary part of the coefficients a j of the convection term for large values of |x|. This condition is known to produce a unique solution with Gevrey regularity of index s ≥ 1 and loss of an infinite number of derivatives with respect to the data for every s ≤ 1 1 - σ . In this paper, we consider the case s > 1 1 - σ , where, in general, Gevrey ill-posedness holds. We explain how the space where a unique solution exists depends on the decay and regularity of an initial data in H m , m ≥ 0 . As a by-product, we show that a decay condition on data in H m produces a solution with (at least locally) the same regularity as the data, but with an expected different behavior as | x | → ∞ . [ABSTRACT FROM AUTHOR]

This paper deals with existence and regularity in variational problems related to partial differential equations and systems - both in the elliptic and in the parabolic contexts - and to calculus of variations, under general and p,q− growth conditions. The manuscript is dedicated to my friend and colleague Patrizia Pucci, with great esteem and sympathy. [ABSTRACT FROM AUTHOR]

This paper investigates the longtime behavior of delayed 3D Navier-Stokes equations in terms of attractors. The study will strongly rely on the investigation of the linearized Navier-Stokes system, and the relationship between the discrete dynamical flow for the linearized system and the continuous flow associated to the original system. Assuming the viscosity to be sufficiently large, there exists a unique attractor for the delayed 3D Navier-Stokes equations. Moreover, the attractor reduces to a singleton set. [ABSTRACT FROM AUTHOR]

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.

We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.

This article presents the maximal regularity approach to the primitive equations. It is proved that the 3D primitive equations on cylindrical domains admit a unique global strong solution for initial data lying in the critical solonoidal Besov space Bpq2/p for p,q∈(1,∞) with 1/p+1/q≤1. This solution regularize instantaneously and becomes even real analytic for t>0. [ABSTRACT FROM AUTHOR]

We study the interrelation between the limit L p (Ω) -Sobolev regularity s ¯ p of (classes of) functions on bounded Lipschitz domains Ω ⊆ R d , d ≥ 2 , and the limit regularity α ¯ p within the corresponding adaptivity scale of Besov spaces B τ , τ α (Ω) , where 1 / τ = α / d + 1 / p and α > 0 ( p > 1 fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N-term approximation. We show how additional information on the Besov or Triebel–Lizorkin regularity may be used to deduce upper bounds for α ¯ p in terms of s ¯ p simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton et al. (in: De Carli and Milman (ed) Interpolation theory and applications, American Mathematical Society, Providence, 2007). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp. [ABSTRACT FROM AUTHOR]