The weak well-posedness results of the strongly damped linear wave equation and of the non linear Westervelt equation with homogeneous Dirichlet boundary conditions are proved on arbitrary three dimensional domains or any two dimensional domains which can be obtained by a limit of NTA domains caractarized by the same geometrical constants. The two dimensional result is obtained thanks to the Mosco convergence of the functionals corresponding to the weak formulations for the Westervelt equation with the homogeneous Dirichlet boundary condition. The non homogeneous Dirichlet condition is also treated in the class of admissible domains composed on Sobolev extension domains of $\mathbb{R}^n$ with a $d$-set boundary $n-1\leq d<n$ preserving Markov's local inequality.The obtained Mosco convergence also alows to approximate the solution of the Westervelt equation on an arbitrary domain by solutions on a converging sequence of domains without additional conditions on their boundary regularity in $\mathbb{R}^3$, or on a converging sequence of NTA domains in $\mathbb{R}^2$.
The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in the largest natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in R^2 or R^3 we also validate the approximation of the solution of the Wester-velt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.
In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincaré-Steklov operator to $d$-set boundaries, $n-2< d<n$, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of $n$-sets. The results are obtained thanks to a generalization of the conti…
We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrödinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl-Titchmarsh $M$-functions and Donoghue-type $M$-functions corresponding to closed extensions of symmetric operators belong to it. The principal purpose of this paper is to prove index formulas that relate the difference of the…
In this paper, we extend the related notions of Dirichlet quasiminimizer, $ω-$minimizer and almost minimizer to the framework of multiple-valued functions in the sense of Almgren and prove Holder regularity results. We also give examples of those minimizers with various branch sets.
We consider the Hölder continuity for the Dirichlet problem at the boundary. Almgren introduced the multivalued; Q-valued functions for studying regularity of minimal surfaces in higher codimension. The Hölder continuity in the interior for Dirichlet minimizers is an outcome of Almgren's original theory, to which C. De Lellis and E.N. Spadaro's work have given a simpler alternative approach. We extend the Hölder regularity for Dirichlet minimizing Q-valued functions up to the boundary assuming enough regularity of the domain and Hölder regularity of the boundary data with exponent >1/2.
In this paper, our goal is to define a measure valued solution of compressible Navier--Stokes--Fourier system for a heat conducting fluid with Dirichlet boundary condition for temperature in a bounded domain. The definition is based on the weak formulation of entropy inequality and ballistic energy inequality. Moreover, we obtain the weak(measure valued)-strong uniqueness property of this solution with the help of relative energy.
This paper discusses the frequency function of multiple-valued Dirichlet minimizing functions in the special case when the domain and range are both two dimensional. It shows that the frequency function must be of value k/2 for some nonnegative integer k. Futhermore, by looking at the blowing-up functions, we characterize the local behavior of the original Dirichlet minimizing function.
We present two new proofs of the exchange theorem for the Laplace transformation of vector-valued distributions. We then derive an explicit solution to the Dirichlet problem of the polyharmonic operator in a half-space. Finally, we obtain explicit solutions to Cauchy-Dirichlet problems of iterated wave- and Klein-Gordon-operators in half-spaces.
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, the tools developed by D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste in the paper "Sobolev inequalities in disguise" allow us to obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. The latter depend not only on th…
Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class $W^{1,p}$, $1 < p < \infty$. If $f$ almost minimizes its $p$ Dirichlet energy then $f$ is Hölder continuous. If $p=2$ and $f$ is squeeze and squash stationary then $f$ is in VMO.
This paper discusses the regularity of multiple-valued Dirichlet minimizing maps into the sphere. It shows that even at branched point, as long as the normalized energy is small enough, we have the energy decay estimate. Combined with the previous work by Chun-Chi Lin, we get our first estimate that m-2 dimensional Hausdorff measure of singular set is zero. Furthermore, by looking at the tangent map and using dimension reduction argument, we show that the singular set is at least of codimension 3.
Inhomogeneous Scaling Function and Heat Kernel Estimates on Fractals Satisfying Some Resistance Conditions
School of Mathematics and Statistics, Beijing Technology and Business University（北京工商大学数学与统计学院）；Center for Applied Mathematics and KL-AAGDM, Tianjin University（天津大学应用数学中心）
With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isope…
Some existence results for a parametric Dirichlet problem defined on the Sierpiński fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution.
In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show that the same upper bound holds for the dimension of the singular set of an area minimizing $n$-dimensional rectifiable current of arbitrary codimension. In either case, the dimension bound is sharp. We develop estimates to study the asymptotic behaviour of a multi-valued Dirichle…
In this paper we establish commmutator estimates for the Dirichlet-to-Neumann Map associated to a divergence form elliptic operator in the upper half-space $\mathbb{R}^{n+1}_+:=\{(x,t)\in \mathbb{R}^n \times (0,\infty)\}$, with uniformly complex elliptic, $L^{\infty}$, $t$-independent coefficients. By a standard pull-back mechanism, these results extend corresponding results of Kenig, Lin and Shen for the Laplacian in a Lipschitz domain, which have application to the theory of homogenization.
We prove results on complex interpolation of vector-valued Sobolev spaces over the half-line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space-valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.
We consider the impedance tomography problem in the plane. Using Bukhgeim's scattering data for the Dirac problem, we prove that the conductivity is uniquely determined by the Dirichlet-to-Neuman map
The paper surveys some recent results concerning vector analysis on fractals. We start with a local regular Dirichlet form and use the framework of 1-forms and derivations introduced by Cipriani and Sauvageot to set up some elements of a related vector analysis in weak and non-local formulation. This allows to study various scalar and vector valued linear and non-linear partial differential equations on fractals that had not been accessible before. Subsequently a stronger (localized, pointwise or fiberwise) version of this vector analysis can be developed, which is related to previous work of …
We consider time-changed Brownian motions on random Koch (pre-fractal and fractal) domains where the time change is given by the inverse to a subordinator. In particular, we study the fractional Cauchy problem with Robin condition on the pre-fractal boundary obtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary conditions on the fractal domain.
We study a parabolic Ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain. We prove that under standard conditions a related Cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial Ventsell problem. As a second result we prove that functions that are intrinsically Lipschitz on the snowflake boundary admit Euclidean Lipschitz extensions to the closure of the entire domain. Our methods combine the fractal membrane analysis, the vector …
We prove dispersive estimates for the linear Schrödinger evolution associated to an operator -Δ+ V, where the potential is a signed measure of fractal dimension at least 3/2.
For the Laplacian $Δ$ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann boundary conditions. That is, we construct a symmetric function $G^{(λ)}$ which solves $(λ\mathbb{I} - Δ)^{-1} f(x) = \int G^{(λ)}(x,y) f(y) dμ(y)$. The method is similar to Kigami's construction of the Green kernel in \cite[\S3.5]{Kig01} and is expressed as a sum of scaled and "translated" copies of a certain function $ψ^{(λ)}$ which may be considered as a fundamental solution of the resolvent equation. …
In this article, first we give a general lemma on the existence of regular homeomorphic solutions $f$ with the hydrodynamic normalization $f(z)=z+o(1)$ as $z\to\infty$ to the degenerate Beltrami equations $\overline{\partial}f=μ\,\partial f$ in $\mathbb C$ whose complex coefficients $μ$ have compact supports. On this basis, we establish criteria for existence and representation of regular discrete open solutions for the Dirichlet problem with continuous data to degenerate Beltrami equations in arbitrary simply connected bounded domains $D$ in $\mathbb C$. Moreover, we obtain similar criteria f…
We present a survey of recent results of the functional analysis allowing to solve PDEs in a large class of domains with irregular boundaries. We extend the previously introduced concept of admissible domains with a d-set boundary on the domains with the boundaries on which the measure is not necessarily Ahlfors regular d-measure. This gives a generalization of Rellich-Kondrachov theorem and the compactness of the trace operator, allowing to obtain, as for a regular classical case the unicity/existence of weak solutions of Poisson boundary valued problem with the Robin boundary condition and t…
In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second author shows that two-valued $C^{1, μ}$ harmonic functions on $n$ dimensional domains, which are typically-non-minimizing stationary points of Dirichlet energy, play an essential role in the study of multiplicity 2 branch points of stable codimension 1 rectifiable currents of dim…
We study the Dirichlet-to-Neumann map for the stationary linear equation of elasticity in a bounded domain in R d , d $\ge$ 2, with smooth boundary. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the principal symbol modulo conjugation by unitary matrices.
Consider the elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, c_{kl} \, \partial_l + \sum_{k=1}^d a_k \, \partial_k - \sum_{k=1}^d \partial_k \, b_k + a_0 \] on a bounded connected open set $Ω\subset {\bf R}^d$ with Lipschitz boundary conditions, where $c_{kl} \in L_\infty(Ω,{\bf R})$ and $a_k,b_k,a_0 \in L_\infty(Ω,{\bf C})$, subject to Robin boundary conditions $\partial_νu + β\, {\rm Tr}\, u = 0$, where $β\in L_\infty(\partial Ω, {\bf C})$ is complex valued. Then we show that the kernel of the semigroup generated by $-A$ satisfies Gaussian estimates and Hölder Gaussian estimates. If …
In his big regularity paper, Almgren has proven the regularity theorem for mass-minimizing integral currents. One key step in his paper is to derive the regularity of Dirichlet-minimizing $\mathbf{Q}_{Q}(\mathbb{R}^{n})$-valued functions in the Sobolev space $\mathcal{Y}_{2}(Ω, \mathbf{Q}_{Q} (\mathbb{R}^{n}))$, where the domain $Ω$ is open in $\mathbb{R}^{m}$. In this article, we introduce the class of weakly stationary-harmonic $\mathbf{Q}_{Q} (\mathbb{R}^n)$-valued functions. These functions are the critical points of Dirichlet integral under smooth domain-variations and range-variations. W…
In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content of the set of singular points with multiplicity $Q$.
The main result of the present paper is a statement on existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems on bounded smooth domains with Dirichlet boundary conditions by means of semigroup theory and fixed point arguments. Main ingredients are the definition of a product of a function and a (not too irregular) distribution as well as a corresponding norm estimate. As an application, transport stochastic partial differential equations driven by fractional Brownian…
We establish a new theory of regularity for elliptic complex valued second order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, when the coefficients of the matrix $A$ satisfy a natural algebraic condition, a strengthened version of a condition known in the literature as $L^p$-dissipativity. Precisely, the regularity result is a reverse Hölder condition for $L^p$ averages of solutions on interior balls, and serves as a replacement for the De Giorgi - Nash - Moser regularity of solutions to real-valued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studie…
The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $t$-independent complex bounded measurable coefficients ($t$ being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in $L^{p'}$, subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in $L^p$. Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with $t…
We prove an existence result for a class of Dirichlet boundary value problems with discontinuous nonlinearity and involving a Leray-Lions operator. The proof combines monotonicity methods for elliptic problems, variational inequality techniques and basic tools related to monotone operators.
In the present paper, we consider a non self adjoint hyperbolic operator with a vector field and an electric potential that depend not only on the space variable but also on the time variable. More precisely, we attempt to stably and simultaneously retrieve the real valued velocity field and the real valued potential from the knowledge of Neumann measurements performed on the whole boundary of the domain. We establish in dimension n greater than two, stability estimates for the problem under consideration. Thereafter, by enlarging the set of data we show that the unknown terms can be stably re…
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition on the local mean oscillations of the coefficients of the differential operator and the unit normal to the boundary (which is automatically satisfied if these functions belong to space VMO) guaranteeing that the solution operator associated with this problem is an isomorphism.
Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we study a selfadjoint matrix elliptic second order differential operator $B_{D,\varepsilon}$, $0<\varepsilon\leqslant 1$, with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves lower order terms with unbounded coefficients. The coefficients of $B_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the generalized resolvent $\left(B_{D,\varepsilon}-ζQ_0(\cdot/\varepsilon)\right)^{-1}$,…
We discuss the Dirichlet problem of the quasi-linear elliptic system \begin{eqnarray*} -e^{-f(U)}div(e^{f(U)}\bigtriangledown U)+&{1/2}f'(U)|\bigtriangledown U|^2&=0, {in $Ω$}, & U|_{\partialΩ}&=ϕ. \end{eqnarray*} Here $Ω$ a smooth bounded domain in $R^n$, $f: R^N\to R$ is a smooth function, $U:Ω\to R^N$ is the unknown vector-valued function, $ϕ:\barΩ\to R^N$ is a given vector-valued $C^2$ function, $f'$ is the gradient of the function $f$ with respect to the variable $U$. Such problems arise in population dynamics and Differential Geometry. The difficulty of studying this problem is that this…
We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings mu : Omega -> (P(D), W\_2) defined over a subset Omega of R^p and valued in the space P(D) of probability measures on a compact convex subset D of R^q endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetn…
The $4N$ carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a $4N$-carpet $F$, let $\{F_n\}_{n \geq 0}$ be the natural decreasing sequence of compact pre-fractal approximations with $\cap_nF_n=F$. On each $F_n$, let $\mathcal{E}(u, v) = \int_{F_N} \nabla u \cdot \nabla v \, dx$ be the classical Dirichlet form and $u_n$ be the unique harmonic function on $F_n$ satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by Barlow and Bass (1990…
