Let $G=(V,E)$ be a graph on $n$ vertices, and let $λ_1(L(G))\ge \cdots\ge λ_{n-1}(L(G))\ge λ_n(L(G))=0$ be the eigenvalues of its Laplacian matrix $L(G)$. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k λ_i(L(G)) \le |E|+\binom{k+1}{2}$. Here, we prove the following weak version of Brouwer's conjecture: For every $1\leq k \leq n$, \[ \sum_{i=1}^k λ_i(L(G)) \leq |E|+k^2+15k\log{k}+65k. \] For a graph $G=(V,E)$, we define its partition density $\tildeρ(G)$ as the maximum, over all subgraphs $H$ of $G$, of the ratio between the number of edges of $H$ and the number of vertices in the largest connected component of $H$. Our argument relies on the study of the structure of the graphs $G$ satisfying $\tildeρ(G)< k$. In particular, using a result of Alon, McDiarmid and Reed, we show that every such graph can be decomposed into at most $k+ 15\log{k}+65$ edge-disjoint star forests (that is, forests whose connected components are all isomorphic to stars). In addition, we show that for every graph $G=(V,E)$ and every $1\le k\le |V|$, \[ \sum_{i=1}^k λ_i(L(G)) \leq |E|+k\cdot ν(G) + \left\lfloor\frac{k}{2}\right\rfloor, \] where $ν(G)$ is the maximum size of a matching in $G$.