On the self-adjoint form for the general elliptic PDE

In fact, there exists a necessary and sufficient condition in order for an elliptic PDE of the form eqref{a} to be put in the self-adjoint form eqref{b}. In order to see this, let's first drop Einstein summation convention and show the summations explicitly. Then, formally (i.e. without considering, for the moment, the differentiability requirements of the functions involved) we have: $$ DeclareMathOperator{divg}{nablacdot}begin{split} divg left(sum_{j=1}^n{a}_{ij}(x) frac{partial u(x)}{partial x_j} right) & = sum_{i=1}^nsum_{j=1}^nfrac{partial}{partial x_i}left({a}_{ij}(x) frac{partial u(x)}{partial x_j} right) \ & = sum_{i=1}^nsum_{j=1}^nleft[ frac{partial a_{ij}(x)}{partial x_i } frac{partial u(x)}{partial x_j} + a_{ij}(x) frac{partial^2 u(x)}{partial x_i partial x_j} right] end{split} $$ This implies that, for the two forms eqref{a} and eqref{b} of a PDE to be equivalent it should be $$ sum_{i=1}^n left[sum_{j=1}^n frac{partial a_{ij}(x)}{partial x_i } frac{partial u(x)}{partial x_j} - b_i(x)frac{partial u(x)}{partial x_i } right]=0label{1}tag{1} $$ However, working a bit by using Kronecker's delta $delta_{ij}$, we have $$ begin{split} sum_{i=1}^n left[sum_{j=1}^n frac{partial a_{ij}(x)}{partial x_i } frac{partial u(x)}{partial x_j} - b_i(x)frac{partial u(x)}{partial x_i } right] & = sum_{i=1}^n left[sum_{j=1}^n frac{partial a_{ij}(x)}{partial x_i } frac{partial u(x)}{partial x_j} - delta_{ij} b_i(x)frac{partial u(x)}{partial x_j } right] \ & = sum_{j=1}^n left[sum_{i=1}^n frac{partial a_{ij}(x)}{partial x_i } frac{partial u(x)}{partial x_j} - delta_{ij} b_i(x)frac{partial u(x)}{partial x_j } right] \ & = sum_{j=1}^n left[sum_{i=1}^n frac{partial a_{ij}(x)}{partial x_i } - delta_{ij} b_j(x) right]frac{partial u(x)}{partial x_j } end{split} $$ and, due to the arbitrariness of $frac{partial u(x)}{partial x_i }$ for $i=1,ldots, n$, we finally get the condition $$ left[sum_{i=1}^n frac{partial a_{ij}(x)}{partial x_i } - delta_{ij} b_j(x) right]=0qquadforall j=1,ldots, n label{2}tag{2} $$ which can be given an elegant form by putting $big(a_{ij}(x)big)_{i,j=1,ldots,n}triangleq mathbf A(x)$ and $mathbf b(x)triangleq {(b_1,ldots, b_n)}$: $$ divgmathbf{A}(x)= mathbf b(x)label{2'}tag{2'} $$

Addendum: is it possible to find an equivalent form eqref{b} for a given non self-adjoint PDE eqref{a}? (Follow-up to the comments of Fizikus)

Several answers to the problem of finding an equivalent symmetric form for a given PDE have been given by researchers who investigated the "inverse problem of the calculus of variations": this problem asks, for a differential equation (ordinary or partial, linear or nonlinear), to find a Lagrangian functional such that the give DE is its Euler-Lagrange equation. Since for a self-adjoint DE the solution of this problem is straightforward, the researchers sought to find ways of transforming non self-adjoint DE in self-adjoint ones.
In particular Copson [A1], for equations of type eqref{a} whose coefficient matrix $A(x)$ is a symmetric, constructed a function $Phi(x)$ and a linear partial differential operator $mathscr{L}_{Phi}(x,partial_i)$ such that the equation $$ e^{Phi(x)} big( mathscr{L}_{Phi}(x,partial_i)u(x) - f(x)big)=0 $$ has the self-adjoint form eqref{b}. The Copson construction is explicit: apart from the original paper, it is also described by Filippov ([A2], §11.2 pp. 94-97), who gives also some "symmetrization methods" which work for particular classes or single equations, both linear and non-linear.

Notes

• Formula eqref{2}-eqref{2'} is seen in classical monographs on the theory of elliptic PDEs ([2], chapter 1, §6 p. 9 and [3] chapter I, §6 p. 12) where it is clearly stated that it is a necessary and sufficient condition for selfadjointness, without proof possibly due to the basic character of the result.
• From the calculation above, it can be seen that condition eqref{2}-eqref{2'} is meaningful for weakly differentiable matrix functions $mathbf{A}(x)$ and $mathbf b(x)$. This implies also that all the above calculations are rigorously valid for an elliptic PDE with weakly differentiable coefficients as, for example, bounded differentiable functions.
• Equation eqref{2}-eqref{2'} in the homogeneous case ($mathbf{b}(x)equivmathbf{0}$) was explicitly solved by Bruno Finzi in 1932 (see [1]) for second, third and fourth order symmetric tensors and after by Maria Pastori in 1942 (see [4]) for every general, not necessarily symmetric, $n$-th order tensors. Perhaps their work could be of some interest in calculating "symmetrizing" factors for transforming the form eqref{a} in the eqref{b}.

References

[1] Bruno Finzi, "Integrazione delle equazioni indefinite della meccanica dei sistemi continui", (Italian), Atti della Accademia Nazionale dei Lincei, Rendiconti, VI Serie 19, pp. 620-623 (1934), JFM 60.0708.02.

[2] Carlo Miranda, Equazioni alle derivate parziali di tipo ellittico, (Italian), Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Heft, Berlin-Göttingen-Heidelberg: Springer-Verlag pp. VIII+222 (1955), MR0087853, Zbl 0065.08503.

[3] Carlo Miranda (1970) [1955], Partial Differential Equations of Elliptic Type, Ergebnisse der Mathematik und ihrer Grenzgebiete – 2 Folge, Band 2, translated by Motteler, Zane C. (2nd Revised ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. XII+370, doi:10.1007/978-3-642-87773-5, ISBN 978-3-540-04804-6, MR 0284700, Zbl 0198.14101.

[4] Maria Pastori, "Integrale generale dell’equazione $operatorname{div}mathsf T=0$ negli spazi euclidei", (Italian) Rendiconti di Matematica e delle Sue Applicazioni, V Serie, 3, pp. 106-112 (1942), MR0018968 Zbl 0027.13302.

Addendum references

[A1] Edward Thomas Copson, "Partial differential equations and the calculus of variations" Proceedings of the Royal Society of Edinburgh 46, 126-135 (1926), JFM 52.0509.01.

[A2] Vladimir Mikhailovich Filippov,, Variational principles for nonpotential operators. With an appendix by the author and V. M. Savchin, Transl. from the Russian by J. R. Schulenberger. Transl. ed. by Ben Silver, Translations of Mathematical Monographs, 77. Providence, RI: American Mathematical Society (AMS). pp. xiii+239 (1989), ISBN: 0-8218-4529-2. MR1013998, ZBL0682.35006.