Nonlinear Dirichlet Forms: Theory & Applications

• Nonlinear Dirichlet forms are convex, lower semicontinuous functionals on Banach spaces that generalize classical bilinear Dirichlet forms through contraction inequalities.
• They provide a unified framework for studying nonlinear evolution equations, p-Laplace and Cheeger energies, and potential theory on nonsmooth or non-Riemannian spaces.
• Key properties like normal contraction, energy space definitions, and modular inequalities facilitate the analysis of Sobolev embeddings and criticality in nonlinear settings.

A nonlinear Dirichlet form is a convex, lower semicontinuous functional on a Banach (typically Hilbert) space, generalizing the classical bilinear Dirichlet form framework to accommodate nonlinearity arising from variational integrals, such as the $p$-energy and Cheeger energies. Such forms provide a unified approach to nonlinear evolution equations, nonlinear potential theory, and the analysis on spaces with non-Riemannian or nonsmooth structure.

1. Definition and Fundamental Properties

Let $V$ be a real topological vector space, with primary interest in $V = L^2(X, m)$ for $(X, m)$ a topological measure space. A functional $E: V \to [0, \infty]$ is called a nonlinear Dirichlet form if it is convex, lower semicontinuous, densely defined, and satisfies a system of contraction inequalities reminiscent of the Beurling–Deny criteria. Specifically, for all $u, v$ in the domain and all $\alpha \ge 0$: $E(u \vee v) + E(u \wedge v) \le E(u) + E(v),$

$$E\big(H_\alpha(u, v)\big) + E\big(H_\alpha(v, u)\big) \le E(u) + E(v),$$

where $H_\alpha(u, v)$ is a piecewise-defined truncation operator. Equivalently, for any normal contraction $C: \mathbb{R} \to \mathbb{R}$ (i.e., $C(0)=0$, $|C(x)-C(y)|\le|x-y|$), one has: $$E(f + Cg) + E(f - Cg) \le E(f + g) + E(f - g) \quad \forall f,g \in V.$$ The subdifferential $\partial E$ generates a nonlinear, order-preserving, contractive semigroup on $L^2$ (i.e., $L^p$-contractions for all $1\le p\le\infty$ and preservation of order) (Puchert, 6 Feb 2025, Claus, 2020, Brigati, 2023, Chill et al., 17 Jan 2026).

2. Energy (Dirichlet) Spaces and Banach Lattice Structure

Given a nonlinear Dirichlet form $E$ as above, the associated energy (Dirichlet) space is

$$\mathcal{D} = \left\{ u \in L^2(X, m): \|u\|_{\mathcal{D}} < \infty \right\}, \quad \|u\|_{\mathcal{D}} = \inf \left\{ \lambda > 0 : \|u / \lambda\|_{L^2}^2 + E(u/\lambda) \le 1 \right\}.$$

This norm is lower semicontinuous and induces a Banach space structure. The space is a Banach lattice: if $u, v \in \mathcal{D}$, then $u \wedge v$, $u \vee v \in \mathcal{D}$, and lattice operations are continuous if $E$ is quasilinear, i.e., its domain is a linear subspace. For even forms ($E(-u)=E(u)$), $(\mathcal{D}, \|\cdot\|_{\mathcal{D}})$ forms a Banach lattice (Claus, 2020, Chill et al., 17 Jan 2026).

3. Beurling–Deny Criteria and Normal Contraction Properties

Nonlinear Dirichlet forms are characterized by generalized Beurling–Deny criteria, reducing to contraction properties under normal contractions. These include

• the modular contraction property: $E(\varphi \circ u) \le E(u)$ for any $1$-Lipschitz $\varphi$ with $\varphi(0) = 0$,
• two-point and truncated contraction relations, encapsulating both the Markovian and order-preserving attributes necessary for sub-Markovian evolution and potential theory (Puchert, 6 Feb 2025, Brigati, 2023).

In the quadratic (bilinear) case, these criteria recover the classical Markov property and the structure theory of linear Dirichlet forms. For the nonlinear case, the contraction property alone suffices to build the theory and ensure the well-posedness of associated nonlinear evolution and potential equations.

4. Capacity, Quasicontinuity, and Potential Theory

Associated to every (symmetric) nonlinear Dirichlet form is a notion of capacity: $$\mathrm{Cap}(A) = \inf \{ \|u\|_{\mathcal{D}} : u \in \mathcal{D},\ u \ge 1 \text{ on some open } U \supset A \}.$$ Capacity is monotone, countably subadditive, inner regular on compacts, and null sets for capacity are null for $m$. Functions in $\mathcal{D}$ admit quasicontinuous representatives: for every $f \in \mathcal{D}$ and $\varepsilon > 0$, there exists open $O$ with $\mathrm{Cap}(O) \leq \varepsilon$ such that $f$ is continuous on $O^c$.

Potential theory for nonlinear Dirichlet forms extends the existence of equilibrium potentials and the structure of exceptional sets (polar/null capacity sets, nests) as in the bilinear framework (Claus, 2020, Schmidt et al., 30 Jan 2025, Beznea et al., 2023).

5. Examples: $p$-Laplace, Cheeger, Variable Exponents, and Quasiregular Maps

• $p$-Laplace type energies: For a strongly local, regular Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ with carré du champ $\Gamma$, define for $p>1$:

$$\mathcal{E}^p(u, v) = \int_X \Gamma(u)^{\frac{p-2}{2}} \Gamma(u, v)\, dm,\quad \mathcal{D}_p = \{ u \in \mathcal{D}(\mathcal{E}) \cap L^p : \Gamma(u)^{1/2} \in L^p \}.$$

This generalizes to the usual $p$-Laplacian in Euclidean domains and to Dirichlet forms associated to quasiregular mappings with the equilibrium and harmonicity theory carried over (Beznea et al., 2023).

• Cheeger energy: On metric measure spaces, Cheeger's energy $\mathrm{Ch}(u)$ is convex, lower semi-continuous, and $2$-homogeneous but typically not quadratic. The corresponding evolution is described by nonlinear heat flow (Brigati, 2023).
• Variable exponent energies: Energies of the form $$\mathcal{E}(u) = \int_M \frac{1}{p(x)}|\nabla u|^{p(x)} d\mu$$ lead to a reflexive nonlinear Dirichlet form structure, under mild boundedness and growth conditions, generalizing the criticality theory (Schmidt et al., 30 Jan 2025).

6. Extensions: Criticality, Extended Spaces, and Sobolev/Isocapacitary Inequalities

The extended Dirichlet space $M(\mathcal{E}_e)$, endowed with the Luxemburg seminorm, is central to potential theory and recurrence/transience classification for nonlinear Dirichlet forms:

• Subcriticality: existence of a Hardy weight $w>0$ with $\int |f|w \le \|f\|_L$, equivalently completeness of $M(\mathcal{E}_e)$.
• Criticality: existence of a null sequence $e_n \uparrow 1$ with $\|e_n\|_L\to0$; triviality of the extended space modulo constants. These criteria parallel classical results for linear forms and extend to the nonlinear context (Schmidt et al., 30 Jan 2025).

Sobolev inequalities for nonlinear Dirichlet forms hold in the form

$\|u\|_{L^q} \leq C\, \|u\|_{\mathfrak{D}},$

and are equivalent to isocapacitary inequalities,

$m(A)^{1/q} \leq C'\, \mathrm{Cap}(A),$

for all measurable sets $A$, providing sharp embeddings and regularization for nonlinear semigroups (Chill et al., 17 Jan 2026).

7. Semigroup Domination and Zero-Order Perturbations

A nonlinear Dirichlet form $E$ generates a nonlinear, order-preserving, contractive semigroup $S_E(t)$ on $L^2$. If a second functional $F$ has its associated semigroup $S_F(t)$ dominated by $S_E(t)$ ($|S_F(t)u|\le S_E(t)|u|$), $F$ is of the form $F(u) = E(u) + \int_X B(x,u(x))\, d\mu(x)$ for a pointwise, nonnegative, lower semicontinuous, bi-monotone kernel $B$, corresponding, e.g., to Robin boundary conditions in PDEs. This generalizes the domination/interpolation theory for boundary-value problems to highly nonlinear contexts (Chill et al., 2023).

---

• (Claus, 2020) "Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting"
• (Brigati, 2023) "Nonlinear Dirichlet forms, energy spaces, and calculus rules"
• (Puchert, 6 Feb 2025) "Nonlinear Beurling-Deny criteria"
• (Beznea et al., 2023) "Nonlinear Dirichlet forms associated with quasiregular mappings"
• (Schmidt et al., 30 Jan 2025) "The extended Dirichlet space and criticality theory for nonlinear Dirichlet forms"
• (Chill et al., 2023) "Domination of nonlinear semigroups generated by regular, local Dirichlet forms"
• (Chill et al., 17 Jan 2026) "Sobolev inequalities for nonlinear Dirichlet forms"