A New Boundary Harnack Principle (Equations with Right Hand Side)
Abstract
We introduce a new boundary Harnack principle in Lipschitz domains for equations with a right hand side. Our approach, which uses comparisons and blowups, will adapt to more general domains as well as other types of operators. We prove the principle for divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.
1 Introduction
1.1 Background
 (1)
In the first example the domain has a sharp corner at the boundary.
 (2)
In the second example \(\Delta x_1^2 =2\) is both positive and too large.
The difference between this example and the first example above is that the cone is wider. The question that naturally arises is: can such behaviour be structured through a general statement, and if so what are the conditions for such a boundary Harnack principle?
Let us also give a simple example where a subsolution controls a solution in the domain \(D = \{x: x_1 >0\}\). We let \(u=x_1\) and \(v=(\varepsilon /2)x_1^2 + x_1\). Although v is subharmonic, it will control the harmonic solution u. In this example \(\Delta v\) is small and \(v(e_1)\) is large. These are the two conditions that will in general guarantee a subsolution may control a solution.
1.2 Main Result
Our main result is the following:
Theorem 1.1
Remark 1.2
We prove Theorem 1.1 for the Laplace operator, however the result will also be true for divergence form operators as given in Theorem 4.7. The methods in Lemmas 3.12– 3.14 will also apply to divergence form operators to show that the quotient is not only bounded as stated in Theorem 4.7 but also Hölder continuous up to the boundary.
For instructional reasons and for the benefit of the nonexpert reader, we first state and prove the theorem inside a cone; see Theorem 2.1. The proof in this case presents the main ideas for the general case. Next, we do the same for a Lipschitz domain; see Theorem 3.10. We then apply the ideas to divergence operators, see Theorem 4.7. Our result holds even when including zeroorder terms. This result appears to be new even when considering the standard boundary Harnack principle without right hand side, that is for solutions rather than supersolutions.
In proving our boundary Harnack principle, we will often utilize the boundary Harnack principle (without right hand side) for two nonnegative solutions. To avoid confusion we will henceforth refer to this as the “standard boundary Harnack principle”.
1.3 Related Results
Another related result is found in [11], where it is shown that a superharmonic function is comparable to the first eigenfunction for a domain in \(\mathbb {R}^2\) with finitely many corners and with an interior cone condition.
1.4 Applications
We present two applications of our boundary Harnack principle: to the HeleShaw flow and to the obstacle problem.
1.4.1 HeleShaw Flow
By a barrier argument (see [10]) one can show that in two dimensions corners with angle smaller than or equal to \(\pi /2\) will not get wet; an analogous result for higher dimensions is a consequence of Theorem 2.5 and 2.6 in this paper. Now the question is what can happen when the angle of the corner is larger than \(\pi /2\); will the liquid reach such a corner? The answer to this question is yes; see [10].
1.4.2 Obstacle Problem
One can actually show that the Lipschitz norm can be taken as small as one wishes by taking the neighbourhood of z smaller. The proof for (nonuniform) Lipschitz regularity is actually much simpler than proving uniform regularity. We refer to [9] for background and details as well as other related original references.
1.5 Future Directions
It seems plausible that the results presented in this paper can be generalized to other operators, as well as more complicated domains. Here we have chosen to treat the problem in Lipschitz domains only. In the final section we consider second order elliptic equations of divergence form. The coefficients are variable and assumed to be only bounded and measurable.
Key elements of our approach are the standard boundary Harnack principle, barrier arguments, as well as scaling and blowup invariance. Since our approach is indirect and uses scalings, the core idea is to look at nonnegative solutions on global domains. The technical difficulties that seem to arise for generaliztions of our result concern the invariance of the domains in scaling.
The methods presented here should also work to prove a boundary Harnack principle for the positivity set of a solution to the thin obstacle problem as long as it is assumed a priori that the free boundary is Lipschitz. Then we may argue in a similar way as above for the thin obstacle problem, with equations having Lipschitz right hand side; see [1] in combination with our result.
2 Boundary Harnack in Cones
Theorem 2.1
Theorem 2.1 is a boundary Harnack principle, but with a right hand side. Clearly, a harmonic solution will control a subsolution. The significance of Theorem 2.1 is that a harmonic solution can control a supersolution, and that the allowed behavior for the right hand side depends on the opening of the cone or more explicitly on \(\alpha _1\). When the opening of the cone is large (so that \(\alpha _1\) is small), then negative values for \(\gamma \) are allowed, and the right hand side can have singular behavior near the boundary. When the opening of the cone is small (so that \(\alpha _1\) is large), then \(\gamma \) must be positive and large, so that the right hand side must decay as it approaches the boundary.
In order to prove Theorem 2.1, we will need the following convergence result:
Lemma 2.2
Proof
An alternate proof of the above lemma, for a more general domain and more general operator, is given in the proof of Lemma 4.5.
We now give a proof of the main theorem in this Section.
Proof of Theorem 2.1
As a corollary we are able to bound a supersolution by a subsolution as long as the right hand side is small enough and the subharmonic solution has a large enough height.
Corollary 2.3
Proof
We also state here another result which will be needed later in the proof of Lemma 3.14:
Corollary 2.4
Proof
We now show that the assumption that \(2\alpha _1 + \gamma >0\) is essential. We first consider the easier case when \(2\alpha _1+\gamma <0\), and show that Theorem 2.1 cannot possibly hold. For clarity of exposition we restrict the analysis to the case when \(\gamma =0\), so that the right hand side is constant.
Theorem 2.5
Proof
We can show that Theorem 2.1 is sharp by considering the critical case when \(2\alpha _1 =0\). For when dimension \(n=2\) this result was shown in [10].
Theorem 2.6
Let \({\mathcal {C}}\) be a cone in \({\mathbb {R}}^n\) with \(\alpha _1=2\). Then the boundary Harnack principle with right hand side does not hold.
Proof
3 Boundary Harnack in Lipschitz Domains

We will employ compactness methods and thus need a convergence result provided by Lemma 3.2.

We will need to bound the behavior of a nonnegative harmonic function at the boundary from above and below which is given in Lemma 3.3.

We will need a Liouville type result which is given in Lemma 3.6.

We will then adapt the proof of Theorem 2.1 (again using compactness techniques) to obtain the proof of Theorem 3.10.
Lemma 3.1
We give later a proof of a more general version of this lemma; see Lemma 4.4 in Section 4.
Lemma 3.2
We give later a proof of the more general Lemma 3.2 in Section 4 that implies Lemma 3.2.
Lemma 3.3
Proof
Corollary 3.4
Proof
Remark 3.5
Lemma 3.6
Let \(u,v \in \mathcal {S}(D_{L,\infty })\) with \(u,v \ge 0\), then \(u = cv\) for some constant \(c\ge 0\).
Proof
Lemma 3.7
Remark 3.8
The significance of Lemma 3.7 is that we do not require \(v \ge 0\).
Proof
Let \(w=C_1uv\ge 0\). Then from Lemma 3.6 we have that \(w=cu\) for some constant c, so that \(v=(C_1c)u\). \(\quad \square \)
We will need an improvement over the previous lemma.
Lemma 3.9
In Section 4 we give a proof of a more general result in Theorem 4.2.
Theorem 3.10
Proof
For simplicity we assume \(\Delta u=0\) and \(\Delta v \le 0\) in \(D_{L,2}\) and show that \(v \le Cu\) in \(B_{1/2}\). Once this has been shown, the situation when u and v satisfy (3.3) is proven exactly as in Corollary 2.3. We also initially prove the theorem for a fixed \(x^0 \in {\mathcal {C}}_M \cap \partial B_{1/2}\).
An interior Harnack inequality with a Harnack chain will also give the result for \(x^0 \in B_{1/2}\cap D_L\) and the constant C depending on dist\((x^0,\partial D_L)\). \(\quad \square \)
Remark 3.11
If one does not assume that \(B_1 \cap \{x_n > 1/4\} \subseteq D_L\), then one may modify the proof at the expense that (3.4) holds for \(x \in B_r\) with \(r \le \min \{(2L)^{1},1/2\}\).
To show that the quotient is not only bounded but also Hölder continuous up to the boundary, we start with the following Lemma:
Lemma 3.12
Proof
This proves the Hölder estimate of u / v up to the boundary along rays in the \(e_n\) direction. The quotient u / v is Hölder continuous in the interior from the usual interior estimates for u and v. This concludes the proof. \(\quad \square \)
Lemma 3.13
Proof
For the application in Section 1.4.2 we actually need to consider the quotient v / u where v is not required to have a sign and \(\Delta v \le M\). This is given in Theorem 1.1. We will need a decay rate for v near points \(x \in \partial D \cap \partial \{v>0\} \cap \partial \{v<0\}\).
Lemma 3.14
Proof
Now we give the proof of Theorem 1.1.
Proof of Theorem 1.1
4 SecondOrder Elliptic Operators
From the forthcoming Lemma 4.1, it will follow that if \(u \ge 0\) and \(u \in \mathcal {S_{\mathcal {L}}}({\mathcal {C}}_L,\infty )\), then u is unique up to multiplicative constant and we again denote u by \(u_L\); however, \(u_L\) will not necessarily be homogeneous. We recall that \({\mathcal {C}}_L\) is defined although not convex when \(L<0\). To emphasize when \(L<0\), we again write \({\mathcal {C}}_{(L)}\) and \(u_{(L)}\) when \(u \in \mathcal {S_{\mathcal {L}}}({\mathcal {C}}_{(L)})\). We will follow the same outline as in Section 3.
In Section 3 we utilized the standard boundary Harnack principle. Since the standard boundary Harnack principle is unavailable when considering the zeroorder term c(x), we prove the next two Lemmas under the situation \(b^i, c \equiv 0\).
Lemma 4.1
Proof
When \(b^i, c \equiv 0\), there is a standard Boundary Harnack principle for divergence form equations [2]; therefore, the proof of Lemma 3.6 holds in this situation, and so the proof of Lemma 3.7 also holds as well. \(\quad \square \)
Theorem 4.2
Remark 4.3
Proof
For the remainder of the section we no longer assume that the lower order terms are zero.
Lemma 4.4
Proof
Since \(u \ge 0\) and \(\mathcal {L} u \le 0\) and \(u \in \mathcal {S_{\mathcal {L}}}(D_{L,R},d^{\gamma })\), this is an application of the interior weak Harnack inequality as well as uniform Hölder continuity up to the boundary, see [7]. \(\quad \square \)
We now state the analogue of Lemma 3.2.
Lemma 4.5
Proof
This is an application of uniform Hölder continuity up to the boundary, see [7]. \(\quad \square \)
Lemma 4.6
Proof
With the previous result, the proof of Theorem 4.7 proceeds exactly as in the case of Theorem 3.10.
Theorem 4.7
Proof
Footnotes
 1.
The boundary Harnack Principle holds in very general domains, such as NTA domains, and uniform domains. It also holds for solutions to a large class of elliptic equations.
 2.
One can actually prove the failure of boundary Harnack between harmonic and superharmonic functions, in the first quadrant. This is illustrated in Example 1.2 in [10], in terms of free boundary problems.
 3.
This was done for constant h in [1] using the standard boundary Harnak principle for Lipschitz domains.
 4.
Actually this conclusion is part of proving the Lipschitz regularity of the free boundary; see [3]
 5.
The level surface is smooth since \(v_{e_1} > 0\) there.
 6.
See for example [9].
 7.
The assumption that \(\partial {\mathcal {C}}\cap \mathbb {S}^{n1}\) is an \((n2)\)dimensional manifold of class \(C^{1,\alpha }\) is not necessary. As we will see in Section 3, \(\partial {\mathcal {C}}\cap \mathbb {S}^{n1}\) may be a Lipshitz manifold provided the Lipschitz constant is small enough depending on \(\gamma \), that appears in (2.2).
Notes
Acknowledgements
Open access funding provided by Royal Institute of Technology. The authors thank the referee for helpful suggestions, such as that to include the case when the right hand side is positive but small.
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