Hopf Type Estimate in Nonlocal PDEs

• Hopf Type Estimate is a quantitative result that extends the classical Hopf lemma to nonlocal and fractional PDEs using anti-symmetry and decay estimates.
• It provides rigorous boundary gradient estimates which are pivotal for proving symmetry via the method of moving planes and maximum principles in integro-differential setups.
• The estimate also quantifies topological invariants in fractional Sobolev spaces, linking analytic bounds with homotopy degrees through precise fractional seminorm controls.

A Hopf type estimate refers to a broad class of results—most notably, boundary gradient estimates analogous to the classical Hopf lemma in elliptic PDE theory, but extended to nonlocal, fractional, or integro-differential settings, as well as sharp quantitative estimates for homotopy invariants in fractional Sobolev spaces. The modern theory encompasses boundary behavior of solutions to nonlocal equations, regularity and symmetry methods, and topological estimates. Key developments include generalized boundary lemmas for fractional Laplacians, integro-differential operators, and estimates of topological invariants (such as the Hopf degree) in fractional function spaces.

1. Classical Hopf Lemma and Nonlocal Extensions

The classical Hopf lemma applies to second-order elliptic operators and states that, under suitable conditions (e.g., $u \geq 0$ in a smooth domain $\Omega$, $u(x^0) = 0$ at a boundary point $x^0 \in \partial\Omega$, and $L u \geq 0$ in $\Omega$), the inward normal derivative at $x^0$ is strictly positive: $\partial_\nu u(x^0) > 0$. This lemma is crucial in strong maximum principle arguments and symmetry results via the method of moving planes.

For nonlocal operators such as the fractional Laplacian,

$$(−Δ)^s u(x) = C_{n,s} \,\mathrm{P.V.}\!\int_{ℝ^n} \frac{u(x) - u(y)}{|x - y|^{n+2s}}\,dy, \quad s \in (0,1),$$

the analogous statement is nontrivial due to the inherent global interactions and lack of pointwise local barriers. Recently, substantial progress has been made formulating and proving Hopf-type boundary estimates for solutions to fractional and integro-differential equations.

2. Fractional Hopf Lemma for Nonlocal Operators

In "A Hopf type lemma for fractional equations" (Li et al., 2017), Li and Chen established a sharp boundary gradient estimate for anti-symmetric functions $w$ solving linear fractional equations in a half-space. Let $\Sigma = \{x \in ℝ^n : x_1 > \lambda\}$ with inward normal $\nu = e_1$. Assume:

• $$w \in C_{\mathrm{loc}}^{1,1}(\Sigma) \cap L^\infty(ℝ^n)$$ and $w(x^\lambda) = -w(x)$ for $\Sigma$-reflection,
• $c: \Sigma \to ℝ$ bounded and continuous to $\partial\Sigma$ with $$\lim_{\mathrm{dist}(x,\partial\Sigma)\to0} c(x)\,\mathrm{dist}(x, \partial\Sigma)^{2s}=0$$,
• $w(x) > 0$ in $\Sigma$,
• $(-Δ)^s w(x) + c(x)w(x) = 0$ in $\Sigma$.

Then, at any $x^0\in \partial\Sigma$,

$$\partial_\nu w(x^0) := \lim_{t\to 0^+} \frac{w(x^0 + t\nu) - w(x^0)}{t} < 0.$$

This is a strict inward gradient bound at the boundary:

$\frac{\partial w}{\partial x_1}(0) < 0,$

if $x^0 = 0$ and $\lambda = 0$ (Li et al., 2017). The proof replaces local barrier arguments with a partitioning of $\Sigma$ and subtle estimates across near-boundary, shell, and far-field regions, leveraging anti-symmetry and decay to control nonlocal effects.

3. Generalized and Quantitative Hopf-type Lemmas for Integro-Differential Operators

A significant generalization is provided in "Hopf type lemmas for subsolutions of integro-differential equations" (Klimsiak et al., 2021). It treats a broad class of Lévy-type integro-differential operators

$$A u(x) = \mathrm{Tr}\left[Q(x) D^2u(x)\right] + b(x)\cdot\nabla u(x) + \int_{ℝ^d} \big[u(x + y) - u(x) - 1_{\{|y|\leq1\}} y\cdot\nabla u(x)\big] N(x,dy).$$

Under strong maximum principle (SMP) and a minorization condition $M(a,\psi,\nu)$ for the semigroup resolvent, a generalized Hopf lemma is derived: For any non-constant weak subsolution $u$,

$$\sup_{y \in \overline{D}^s} u(y) - u(x) \geq A\, \psi(x), \quad \forall x \in D,$$

where $\overline{D}^s$ is the nonlocal closure, and $A>0$ depends on $u$ and the process coefficients. Quantitative versions provide explicit forms when $u$ attains its supremum at the (extended) boundary and under irreducibility conditions, yielding bounds in terms of the gauge function $w_{c,D}(x)$ (Klimsiak et al., 2021).

A crucial corollary is that, for killed semigroups satisfying intrinsic ultracontractivity, the principal Dirichlet eigenfunction $\phi_D$ serves as $\psi$, and strict lower bounds for subsolutions ensue:

$$\sup_{y \in \overline{D}^s} u(y) - u(x) \geq A\, \phi_D(x), \quad x \in D,$$

with $A>0$ depending on $u$.

4. Applications: Symmetry, Maximum Principles, and Moving Planes

Hopf-type boundary estimates play a decisive role in the method of moving planes for symmetry analysis of nonlocal equations. For instance, for solutions to $(-Δ)^s u = f(u)$ in $ℝ^n$ with $u > 0$, defining

$w_\lambda(x) = u(x^\lambda) - u(x),$

one starts with $w_\lambda \geq 0$ deep in the half-space, and decreases $\lambda$ to a critical position $\lambda_0$. The Hopf-type lemma ensures that if a minimum is achieved at the boundary, the normal derivative is strictly negative—yielding a contradiction unless symmetry is enforced, i.e., $w_{\lambda_0} \equiv 0$ (Li et al., 2017). This framework extends to more general nonlocal, possibly nonlinear operators when suitable maximum principles and kernel estimates hold.

5. Hopf-type Estimates for Topological Invariants in Fractional Sobolev Spaces

In "An estimate of the Hopf degree of fractional Sobolev mappings" (Schikorra et al., 2019), Hopf-type bounds are established not for PDE solutions, but for topological functionals arising in fractional Sobolev mapping theory. For smooth maps $f: S^{4n-1} \rightarrow S^{2n}$ with $s \in [1-1/(4n), 1]$, $p = (4n-1)/s$, and fractional seminorm

$$[f]_{W^{s,p}(S^{4n-1})} = \left( \iint_{S^{4n-1} \times S^{4n-1}} \frac{|f(x) - f(y)|^p}{|x - y|^{(4n-1)+sp}}\,dx\,dy \right)^{1/p},$$

the estimate reads:

$$|\deg_H(f)| \leq C(n,s) [f]_{W^{s,p}(S^{4n-1})}^{4n/s},$$

where $\deg_H(f)$ is the Hopf invariant, expressed via the Whitehead integral formula. The analytic heart of the argument is a commutator (Jacobian) estimate relating the Riesz potential of certain pullback forms to the appropriate fractional Sobolev norm. The exponent $4n/s$ is sharp, as demonstrated by explicit families of maps.

6. Connections to Ergodicity, Maximum Principles, and Nonlocal Geometry

The general theory reveals that strong maximum principles, ergodic properties of semigroups, and irreducibility of the resolvent are intimately linked with the validity and quantitative strength of Hopf-type estimates (Klimsiak et al., 2021):

• Minorization conditions (M(a, ψ, ν)) ensure resolvent lower bounds.
• Intrinsic ultracontractivity of semigroups guarantees spectral gap and existence of a strictly positive eigenfunction usable in Hopf-type inequalities.
• Uniform ergodicity is equivalent to the φ_D-Hopf lemma holding for all subsolutions.
• Irreducibility is both necessary and sufficient (under mild conditions) for minorization, bridging stochastic, analytic, and geometric approaches.

The nonlocal nature of the operators mandates working with extended boundaries, nonlocal closures, and functions defined globally or with appropriate decay. In particular, "barrier" arguments must be replaced by careful control of nonlocal interaction terms, partitioning regions in space to distinguish dominant and negligible contributions.

7. Summary Table: Key Hopf-type Estimates

Context/Operator	Main Estimate	Reference
Fractional Laplacian ($(-Δ)^s$)	$\partial_\nu w(x^0) < 0$ at boundary under symmetry	(Li et al., 2017)
Lévy-type integro-differential operator	$u_{\partial^s D} - u(x) \geq A \psi(x)$	(Klimsiak et al., 2021)
Fractional Sobolev mapping $f:S^{4n-1}\to S^{2n}$	$|\deg_H(f)| \leq C [f]_{W^{s,p}}^{4n/s}$	(Schikorra et al., 2019)

These results collectively represent the state-of-the-art in Hopf-type gradient and topological estimates in nonlocal, fractional, and geometric analysis. The field is evolving towards greater unification of analytic, stochastic, and topological methods, with ongoing extensions to nonlinear equations, more general kernels, and further topological invariants.