# Simple existence and uniqueness for second order and linear elliptic PDE

Consider a closed Riemannian manifold $$(M,g)$$ and let $$u \in C^{2,\alpha}(M)$$ be a positive function on $$M$$.

I am interested on the existence of solution for the following problem: given a continuous function $$\psi$$ on $$M$$, when does there exists unique $$v \in C^{2,\alpha}(M)$$ such that $$\Delta v + F(u,\nabla u)v + G(u,\nabla u)g(\nabla u,\nabla v) = \psi$$ where $$F$$ and $$G$$ are smooth functions on their parameters and $$u\in C^{2,\alpha}(M)$$ is fixed.

This should not be a very difficult problem for one reason: this PDE is nothing more than a linear elliptic second order PDE, so it has a well consolidated theory.

The problem is that in general, to find a solution for this problem one needs to ensure that $$\psi$$ is in the orthogonal complement of the dual of the linear and elliptic operator $$P_u : v \mapsto \left(\Delta(\cdot) + F(u,\nabla u)\cdot + G(u,\nabla u)g(\nabla u,\nabla\cdot)\right)v,$$ and in particular, this seems very difficult to compute since the expressions for $$F$$ and $$G$$ are quite complicated.

My question is: does there exists a way of ensuring the existence for solution to my problem in an easy way, i.e, without looking for the kernel of $$P^*_u$$?

I have a particular guess that this can be done, how so?

You are dealing with a linear, second-order elliptic linear partial differential operator (LPDO) $$P_u$$ with $$C^{2,\alpha}$$ coefficients (you have to take $$\alpha\in(0,1)$$) on the left hand side of the equation, whose principal part is the Laplacian $$\Delta$$ on $$(M,g)$$. The standard approach to existence of solutions to such an equation when the right hand side $$\psi\in C^\alpha$$ ($$\psi$$ just continuous will not work, see below) is by combining a priori estimates for $$P_u$$ in Hölder spaces with an abstract method called the method of continuity - the idea is to interpolate between $$\Delta$$ and $$P_u$$ by setting $$P_0=\Delta$$, $$P_1=P_u$$ and $$P_t=(1-t)P_0+tP_1$$ ($$t\in[0,1]$$). The basis of such a method is the following theorem (see e.g. Theorem 5.2, pp. 75 of the book by D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of the Second Order, Springer-Verlag, 1983):
Theorem: Let $$P_0,P_1:B\rightarrow V$$ be bounded linear maps from the Banach space $$B$$ into the normed vector space $$V$$, and $$P_t$$ as above. If there is $$C>0$$ independent of $$t$$ such that $$\|v\|_B\leq C\|P_t v\|_V$$ for all $$t\in[0,1]$$, then $$P_1$$ is surjective if and only if $$P_0$$ is (injectivity of both operators is obvious from the above estimate).
In other words, the solvability of your equation will come from the solvability of the Poisson equation on $$(M,g)$$ with the same right hand side (this, on its turn, is standard). The above estimate is the a priori estimate we need - it usually comes from the so-called Schauder estimates in Hölder spaces (see e.g. the book by Gilbarg and Trudinger cited above in the case $$(M,g)$$ is a compact domain in $$\mathbb{R}^n$$ with smooth boundary).
The problem with assuming $$\psi$$ being just continuous is that there may be not even $$C^2$$ solutions $$v$$ to $$\Delta v=\psi$$ (let alone $$C^{2,\alpha}$$) in this case. Problem 4.9 (a), page 71 of Gilbarg-Trudinger, loc. cit. provides a (counter)example of a continuous, compactly supported function $$f$$ in $$\mathbb{R}^n$$, $$n\geq 2$$, such that $$\Delta v=f$$ fails to have a $$C^2$$ solution $$v$$ in any given open neighborhood of the origin.
• Pedro, thank you very much for such a complete answer. I did not make the calculations yet, but I am confident this will solve my problem, since although $F,G$ are a little bit complicated, I think I can handle it. About $\psi$ being $C^0,$ in my case it is smooth indeed, but I had no idea about the existence of counter examples when it is just continuous. Thank you. – L.F. Cavenaghi Nov 19 '19 at 1:38