Hausdorff Dimension Of Singular Set Research Articles

A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems

We prove partial regularity for minimizers to elasticity type energies with [Formula: see text]-growth, [Formula: see text], in a geometrically linear framework in dimension [Formula: see text]. Therefore, the energies we consider depend on the symmetrized gradient of the displacement field. It is an open problem in such a setting either to establish full regularity or to provide counterexamples. In particular, we give an estimate on the Hausdorff dimension of the potential singular set by proving that is strictly less than [Formula: see text], and actually [Formula: see text] in the autonomous case (full regularity is well-known in dimension [Formula: see text]). The latter result is instrumental to establish existence for the strong formulation of Griffith type models in brittle fracture with nonlinear constitutive relations, accounting for damage and plasticity in space dimensions [Formula: see text] and [Formula: see text].

Partial regularity for the 3D magneto-hydrodynamics system with hyper-dissipation

We prove that for the 3D MHD equations with hyper-dissipations (-Δ)α (1 < α < 5/4) the Hausdorff dimension of singular set at the first blowing up time is at most 5 − 4α, by means of physical and frequency localization, Bony’s paraproduct and Littlewood-Paley theory.

On the Hausdorff Dimension of the Singular Set in Time for Weak Solutions to the Non-stationary Navier–Stokes Equation on Torus

In this note, we investigate the Hausdorff dimension of the possible time singular set of weak solutions to the Navier–Stokes equation on the three dimensional torus under some regularity conditions of Serrin’s type (Arch. RationalMech. Anal., 9, 187–195, 1962). The results in the paper relate the regularity conditions of Serrin’s type to the Hausdorff dimension of the time singular set. More precisely, we prove that if a weak solution u belongs to Lr(0, T; Vα) then the \(\left (1-\frac {r(2\alpha -1)}{4}\right )\)-dimensional Hausdorff measure of the time singular set of u is zero. Here, r is just assumed to be positive. We also establish that if a weak solution u belongs to Lr(0, T; W1, q) then the \(\left (1-\frac {r(2q-3)}{2q}\right )\)-dimensional Hausdorff measure of the time singular set of u is zero. When r = 2, α = 1, or r = 2, q = 2, we recover a result of Leray (Acta Math. 63, 193–248, 1934), Scheffer (Commun. Math. Phys. 55, 97–112, 1977), Foias and Temam (J. Math. Pures Appl. 58, 339–368, 1979), and Temam (Navier–Stokes equations and nonlinear functional analysis. SIAM, Philadelphia, 1995). Our results in some way also relate to the regularity results obtained by Giga (J. Differ. Equ. 62, 186–212, 1986).

Quantitative Stratification and the Regularity of Harmonic Maps and Minimal Currents

AbstractWe provide techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and their consequences) into more effective control. In the present paper, we focus on proving regularity theorems for minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on the volumes of their r‐tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps $f: M^n \rightarrow N^m $ between Riemannian manifolds include a priori bounds in $ W^{1,p} \cap W^{2,{p}/{2}} $ for all p &lt; 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control actually provides much stronger information than that which follows from a bound on the Lp norm of derivatives. Namely, we obtain Lp bounds for the reciprocal of the regularity scale $r_f(x):= \max\{r: \sup_{B_r(x)}r|\nabla f|+r^2|\nabla^2 f|\leq 1\}$. Applications to minimal hypersufaces include a priori Lp bounds for the second fundamental form A for all p &lt; 7. Previously known bounds were for $ p &lt; 4+ \sqrt{{8}/{n}} $ in the smooth immersed stable case. Again, the full theorem is much stronger and yields Lp bounds for the reciprocal of the corresponding regularity scale $r_{|A|}(x):= \max\{r: \sup_{B_r(x)}r|A|\leq 1\}$. In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed Riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key role in each of these arguments is played by the relevant quantitative differentiation theorem. © 2013 Wiley Periodicals, Inc.

On the Hausdorff dimension of singular sets for the Leray-α Navier-Stokes equations with fractional regularization

On the Hausdorff dimension of singular sets for the Leray-α Navier-Stokes equations with fractional regularization

Singular dimension of the solution set of a class of p-Laplace equations

We consider the boundary value problem −Δ p u = F(x), in a bounded open domain Ω ⊆ ℝ N , where F ∈ L p ′(Ω), 1 < p < ∞, p′ = p/(p − 1). Let X(Ω, p) be the set of weak solutions u generated by all right-hand sides F. Define the singular dimension of the solution set as the supremum of Hausdorff dimension of singular sets of solutions in X(Ω, p), and denote it by s-dim X(Ω, p). We show that for p > 2 we have s-dim X(Ω, p) = (N − pp′)+, where r + = max{0, r}. In the proof we exploit among others a regularity result for p-Laplace equations due to J. Simon [Sur des Équations aux Dérivées Partielles Non Linéaires, Thése, Paris, 1977], involving Besov spaces. For 1 < p < 2, an estimate for the singular dimension of the solution set is obtained.

Generating Singularities of Weak Solutions of $p$-Laplace Equations On Fractal Sets

We study $p$-Laplace equations $-\Delta_p u=F(x)$ possessing weak solutions in the Sobolev space $W_0^{; ; 1, p}; ; (\Omega)$, $\Omega\subset\mathbb{; ; R}; ; ^N$, that are singular on prescribed fractal sets having large Hausdorff dimension. With an appropriate choice of $F\in L^{; ; p'}; ; (\Omega)$ the Hausdorff dimension of singular set of the weak solution can be made arbitrarily close to $N-pp'$ if $pp'<N$. For $p=2$, that is, for the classical Laplace equation, the bound $N-4$ is optimal, provided $N\ge4$. Moreovoer, there exist maximally singular solutions, that is, such that the bound is achieved. The proof is obtained via an explicit lower a-priori bound of supersolutions corresponding to special choice of right hand-sides that are singular near a fractal set.

The Hausdorff Dimension of Singular Sets of Properly Discontinuous Groups

The Hausdorff Dimension of Singular Sets of Properly Discontinuous Groups

The Hausdorff dimension of singular sets of properly discontinuous groups in 𝑁-dimensional space

The Hausdorff dimension of singular sets of properly discontinuous groups in 𝑁-dimensional space