A trace theorem for spherical kinetic Sobolev spaces on $ρ$-convex domains

Abstract: Trace theorems are an indispensable tool for the analysis of kinetic equations. They have been established in wide generality by Cessenat and co-workers for radiative transfer and related applications. A variety of trace estimates have been established for the kinetic Fokker-Planck and Kolmogorov equation, typically requiring smoothness of the underlying domain; see the recent survey by Niebel & Valentini. In this work, we prove a new trace estimate for kinetic Sobolev spaces over the sphere for $ρ$-convex domains which, in general, may have a non-smooth boundary. Similar to the work of Cessenat, we use characteristics to obtain trace estimates in weighted trace spaces with explicit constants. For completeness, we also present a density result for the corresponding function spaces on Lipschitz domains.

• The paper establishes a continuous boundary trace for spherical kinetic Sobolev functions on ρ-convex domains, proving an explicit estimate with constant 2(1+4ρ) under the weight τ|s·n|.
• The proof exploits the 2ρ-Lipschitz dependence of characteristic exit times on the spherical direction and controls transport terms through an H^{-1}(S)–H^1(S) dual pairing.
• The paper proves smooth-function density in the graph space on bounded Lipschitz domains, enabling trace extension while identifying weaker weights as the trade-off for allowing non-smooth boundaries.

The paper develops a trace theory for the spherical kinetic Sobolev space

$$H_{hyp} = \{v \in L^2(R;H^1(S)) : s\cdot\nabla_r v \in L^2(R;H^{-1}(S))\},$$

equipped with the graph norm $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$, over a class of domains that may have non-smooth boundaries. The motivation comes from kinetic Fokker–Planck equations, where uniqueness arguments via Lions' representation theorem require traces on the boundary of the phase space, and where the density of smooth functions vanishing on the outflow boundary in the associated graph space remains open (2608.19027).

The class of ρ-convex domains

The central geometric assumption is $\rho$-convexity: a domain $R \subset \mathbb{R}^d$ is $\rho$-convex if $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$ for some radius $\rho>0$ and a set of centers $A$. Such domains are convex and bounded, but may have corners — the Reuleaux triangle (intersection of three balls) is the canonical example. Crucially, the outward normal exists for $\mathcal{H}^{d-1}$-almost every boundary point, and the exit times $\tau_\pm(r,s)$ along characteristics are well defined. The paper's main technical observation is that $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$0-convexity forces directional regularity of the exit times: $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$1 and $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$2, hence $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$3 almost everywhere. The proof exploits the explicit formula for the exit time in a ball and the fact that for a $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$4-convex domain, $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$5; the Lipschitz bound is then obtained by taking infima of the ball-level inequalities. A special case of this lemma appeared previously, but the full statement and proof given here are self-contained.

The trace theorem

The main result states that for every $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$6-convex $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$7 and all $$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$8,

$$\|v\|_{hyp}^2 = \|v\|_{L^2(R;H^1(S))}^2 + \|s\cdot\nabla_r v\|_{L^2(R;H^{-1}(S))}^2$$9

with an explicit constant depending only on $\rho$0. By density of smooth functions, the trace map extends continuously to all of $\rho$1. The proof follows the characteristics method of Cessenat: along a ray, the quantity $\rho$2 satisfies $\rho$3, and integration over the inflow boundary using the standard change-of-variables identity for characteristics reduces the boundary integral to volume terms. The transport term is then controlled in the dual pairing $\rho$4, using the bound $\rho$5 together with Young's inequality. The final constant follows by summing the estimates for $\rho$6 and $\rho$7.

A structural remark is that the weight $\rho$8 is genuinely weaker than the weights $\rho$9 ($R \subset \mathbb{R}^d$0) appearing in trace estimates of Niebel–Valentini, which require $R \subset \mathbb{R}^d$1 boundaries. Indeed, for $R \subset \mathbb{R}^d$2-convex domains one has $R \subset \mathbb{R}^d$3, so $R \subset \mathbb{R}^d$4; the reverse bound fails in general. Thus the paper trades a smaller (weaker) trace norm for a substantial relaxation of boundary regularity, at the cost of an explicit constant that is not claimed to be sharp.

Density of smooth functions

To justify the extension of the trace operator, the paper proves that $R \subset \mathbb{R}^d$5 is dense in $R \subset \mathbb{R}^d$6 for any bounded Lipschitz domain. The core result is for domains star-shaped with respect to a ball $R \subset \mathbb{R}^d$7, established via a three-stage approximation: (i) global dilation $R \subset \mathbb{R}^d$8, which by star-shapedness satisfies $R \subset \mathbb{R}^d$9 with distance at least $\rho$0 from the boundary, and which converges to the identity in the graph norm; (ii) Euclidean mollification in $\rho$1, which commutes with $\rho$2; (iii) fibre-wise application of the heat semigroup $\rho$3 on $\rho$4, which commutes with $\rho$5 and is contractive on $\rho$6. A diagonal sequence yields the approximation. The Lipschitz case follows by a partition-of-unity localization over a finite cover by star-shaped subdomains. This slightly generalizes prior density results for the spherical velocity model, which required $\rho$7 regularity, and the paper notes that the alternative of defining the graph space by completion of smooth functions (as in some earlier work) leaves its relation to the transport graph space implicit — a gap the present approach closes.

Extensions and limitations

The authors indicate that the density argument extends to time-dependent spaces $\rho$8 with $\rho$9, since the space-time exit times $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$0 (respectively $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$1) inherit the $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$2-Lipschitz dependence on $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$3. Two limitations are stated plainly. First, the question whether estimates with the stronger weight $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$4 remain valid on $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$5-convex or other non-smooth domains is open; counterexamples show such estimates fail for general rough boundaries. Second, the directional Lipschitz estimate on $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$6 fails for domains without uniform convexity, so the method does not directly cover polyhedral domains; the authors announce a forthcoming work establishing weighted trace estimates of the form $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$7 with appropriate weights $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$8 for polyhedral domains in dimensions two and three, by different methods.

Conclusion

The paper contributes a trace theorem for spherical kinetic Sobolev spaces with an explicit constant $$\overline{R} = \bigcap_{a\in A} \overline{B(a,\rho)}$$9, valid on $\rho>0$0-convex domains with possibly non-smooth boundaries, together with a density result for Lipschitz domains that underpins its extension by continuity. The weighted trace norm obtained is weaker than those available for smoother domains, and the trade-off between boundary regularity, weight strength, and geometric assumptions on the exit times remains the principal open direction.