Heat dispersion laws in smooth compact manifolds

Abstract: Given a Lipschitz conductor $K$ in the smooth compact Riemannian $2\le n$-manifold $(M,g)$, such a half generic heat dispersion law $$ {\rm H^d}_{p,\varPhi,\varPsi}(K,M)=2^{-1} {\rm H^d}_{Δ_p,\varPhi,\varPsi}(K,M) $$ is not only newly-established via Theorem 1.1 but also deeply-explored through not only Proposition 3.1 (a comparison law for the generic heat dispersion) but also Proposition 3.2 (a recycling law for the quasilinear Laplace-Robin eigenvalue).

• The paper’s central contribution is proving an exact one‐half factor between direct heat dispersion and the p-Laplacian formulation.
• It employs rigorous variational methods in W¹, p spaces to demonstrate unique minimizers and derive Robin-type boundary conditions.
• Geometric comparison results using curvature bounds yield sharp spectral optimization insights for heat conduction in complex geometries.

Heat Dispersion Laws in Smooth Compact Manifolds

Introduction and Motivation

The study addresses mathematical-physical laws governing heat dispersion in smooth compact Riemannian manifolds $(M,g)$ with boundary. The paper situates itself at the intersection of geometric analysis, nonlinear potential theory, and mathematical physics, and generalizes classical results on isoperimetry and capacity to the context of nonlinear heat diffusion, notably via the $p$-Laplacian and related non-Euclidean variational principles.

Previous developments, notably in \cite{Br1,Br2,Br3,BE}, establish fundamental geometric inequalities (e.g., Michael-Simon-Sobolev and isoperimetric) and their links to curvature and boundary behavior. This work extends such frameworks to define and analyze functionals that describe the generic and half-generic heat dispersion for conductors embedded in a manifold, offering new variational principles and comparison results.

The Generic and Half-Generic Heat Dispersion Laws

Let $K\subset M$ be a compact Lipschitz domain designating a conductor at unit temperature, with $M\setminus K$ representing the surrounding insulator. The paper introduces the generic heat dispersion functional for any $p\in[1,\infty)$ and smooth, nonnegative potentials $\varPhi$ (in $M$) and $\varPsi$ (on $\partial M$):

$${\rm H^d}_{p,\varPhi,\varPsi}(K, M) = \inf_{f\in W^{1,p}(M),\, f|_K=1} \left\{ \int_M \left(|\nabla f|^p + \varPhi|f|^p\right)\,d\upsilon_g + \int_{\partial M} \varPsi |f|^p\,d\sigma_g \right\}$$

The functional quantifies the minimal "cost" (in terms of a $p$0-energy plus potential) to sustain unit temperature in $p$1 under the influence of $p$2 and $p$3. The infimum is achieved by a unique minimizer $p$4, as shown via direct method arguments leveraging reflexivity, weak compactness, and Sobolev trace embeddings.

Additionally, a related (quasi)linear operator is defined:

$p$5

where $p$6 is the $p$7-Laplacian and $p$8 is a suitable class of $p$9 extensions of the boundary-value constraint.

The central result (Theorem 1) is the establishment of a half-generic heat dispersion law:

$K\subset M$0

This result provides an exact relationship between the variational energies associated with the direct heat flow ($K\subset M$1-Dirichlet problem) and the quasilinear operator involving the $K\subset M$2-Laplacian and potentials.

Analysis of Variational Characterization

The uniqueness and existence of minimizers are established through the direct method in calculus of variations, using the compactness of $K\subset M$3 and properties of $K\subset M$4 spaces. The Euler-Lagrange equations for the minimizer characterize the weak solution to the boundary-value problem with Robin-type boundary conditions (involving $K\subset M$5), leading to a system of nonlinear PDEs with mixed (Dirichlet/Robin) data.

The proof of the half-generic law exploits delicate regularity arguments and mollification near the boundary of $K\subset M$6, employing specifically designed cutoff functions $K\subset M$7 to circumvent regularity and compatibility obstacles at $K\subset M$8. This allows the use of $K\subset M$9 in the more rigid variational characterization for $M\setminus K$0, closing the gap between the direct and indirect functionals.

The paper further identifies extreme situations:

• For vanishing potentials $M\setminus K$1 and $M\setminus K$2 (purely insulated), the minimal value corresponds to a $M\setminus K$3-capacity with Dirichlet data.
• For diverging potential $M\setminus K$4, the functional diverges, indicating the physical impossibility of thermal conduction under infinite absorption.

Comparison and Recycling Laws

Comparison Law (Proposition 3.2)

Leaning on geometric comparison theory and curvature lower bounds, the authors establish that, under suitable Ricci and mean curvature bounds, the heat dispersion in $M\setminus K$5 is upper bounded by that in a corresponding model space of constant curvature $M\setminus K$6 with equivalent boundary area. This sharp comparison, using explicit construction of test functions derived from the model minimizer, enables isoperimetric-type control of heat dispersion based on curvature data and boundary geometry. Formally:

$M\setminus K$7

where $M\setminus K$8 is a symmetric region in the model manifold determined by equidistance from the boundary.

Eigenvalue Recycling Law (Proposition 3.1)

The authors derive a recycling law for the quasilinear Laplace-Robin ($M\setminus K$9-Laplacian with Robin condition) eigenvalue. By clever use of the variational framework, it is shown that the variational characterization involving the magnitude of residuals from the eigen-PDE exactly recycles the eigenvalue:

$p\in[1,\infty)$0

for both finite and infinite Robin parameter ($p\in[1,\infty)$1 yields the Dirichlet problem).

Furthermore, spectral isoperimetric inequalities for the first Robin or Dirichlet $p\in[1,\infty)$2-Laplacian eigenvalues are derived as corollaries of the heat dispersion comparison, emphasizing the geometric dependence of thermodynamic and spectral characteristics.

Numerical and Analytical Implications

Key sharp results and equivalences are established, notably:

• Exact factor of one-half between direct and $p\in[1,\infty)$3-Laplacian mediated heat dispersion functionals.
• Comparative control of heat dispersion by curvature and boundary conditions, placing the analysis within the field of quantitative isoperimetry.
• Sharp characterization of spectral values (Robin/Dirichlet $p\in[1,\infty)$4-Laplacian eigenvalues) via recycling laws, facilitating exact variational formulations for spectral optimization in geometric settings.

These results not only provide robust mathematical tools for the analysis and optimization of heat flow in complex geometries but also lay foundational groundwork for the study of related variational inequalities and nonlinear PDEs on manifolds.

Theoretical and Practical Outlook

The implications are twofold:

• Mathematical Physics: The exact relationships and comparison principles offer powerful tools for understanding conductivity, insulation, and diffusion in general geometric contexts, with potential applications in thermodynamics, materials science, and geometric PDEs.
• Geometric Analysis and Spectral Theory: The established variational principles and comparison laws renew and generalize classical results connecting geometry, curvature, and spectral invariants, inspiring new avenues in isoperimetric, capacity, and Sobolev inequalities on Riemannian manifolds.

Further developments may include:

• Extending the results to non-compact or singular manifolds.
• Considering anisotropic conductivities or nonlocal operators.
• Investigating the full spectrum of the $p\in[1,\infty)$5-Laplacian under similar variational and geometric constraints.
• Data-driven or computational approaches to optimizing heat dispersion in engineered geometric structures.

Conclusion

The paper systematically develops a deep analytic and geometric framework for understanding heat dispersion governed by the $p\in[1,\infty)$6-Laplacian on smooth compact manifolds. By introducing and analyzing novel energy functionals, proving sharp comparison and recycling laws, and establishing connections with spectral theory, the work contributes significant theoretical tools and results linking geometry, analysis, and mathematical physics. The established principles should have lasting impact both for the theory of nonlinear PDEs on manifolds and for practical problems in geometric heat transfer.

Reference:

"Heat dispersion laws in smooth compact manifolds" (2605.06174)